I've never seen or read anything about youth who were specifically against rock and roll, most sources focus on the moral panic adults displayed-but that doesn't mean there weren't kids who didn't rebel by listening to rock music in the 50s and 60s, however, that doesn't mean that those kids were listening to older music (in fact, since ragtime was largely forgotten by the general public until it was re-popularized in the 1970s by Joshua Rifkin and then the soundtrack of The Sting its a safe bet they didn't listen to Ragtime)-More likely they would have been listening to "safe" pop artists like Neil Sedaka and Paul Anka. Or, maybe they weren't listening to pop-keep in mind that wasn't the only choice. We like to divide history into neat little eras and assume that since the rock and roll era started in the 1950s, that meant Jazz's time in the sun was over-but two of the most important-and most popular jazz albums of all time, Kind of Blue by Miles Davis and Take Five by Dave Brubeck were released in 1959- so plenty of people were still listening to Jazz and not listening to pop or rock at all.

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There's a larger reason the kids weren't listening to older music though- the idea of "Oldies" and "Classic Rock" type stations are a relatively new thing.

Until the 1970s (with the advent of the "Golden Oldies" format, playing music from the 50s and 60s) there wasn't really a radio format that focused on music more than a few years old...Music radio was a vehicle to drive music sales- first of sheet music, then of records, so there wasn't a platform to play the songs of yesteryear-because record companies weren't trying to make money off of those songs, they were plugging today's new big thing.

Off and on since the 1990s, every once in a while there will be a news story about artists trying to get royalties paid out of their radio airplay-because currently the performer (in the US) doesn't receive a royalty payment. This didn't originally happen because the record companies were trying to screw the artist (at least not in that particular way)- the idea in 1950 or 1960 of the version of a song an artist had just recorded still receiving airplay even 5 years, much less 50-70 years later like some songs do now was a complete anathema to everyone involved-it just wasn't done. The chief purpose of radio airplay was to promote the artist's new album and drive sales, it wasn't looked at as a major cash cow yet.

So in short, kids probably weren't listening to music from a few decades ago and lamenting the sad state of modern pop music-and if they were, they weren't doing it on the radio or finding old copies of Duke at Carnegie Hall at the record store to take home and play.

Were there kids lamenting how modern music was terrible, along with their parents? Probably. But they probably weren't listening to 50 year old music while doing so. In other words, its a safe bet no one in the 1950s said this.

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Sources: Jazz by Geoffrey Ward and Ken Burns, Scott Joplin Complete Piano Works, What You Didn;t Know About Radio Royalties, A History of Music Choice in Radio Programming, Rock and Roll and Moral Panics

Edit: Source I forgot: On The Shoulders of Giants by Kareem Abdul-Jabbar

The whole /r/lewronggeneration phenomenon is fascinating, honestly - and I say that as someone who writes a lot of very in-depth, detailed answers about the Beatles borne of a lifetime of listening to them (as well, as a fair bit of reading), despite my not being born until after John Lennon was killed. The phenomenon speaks of a certain level of older music being considered as a sort of 'counter-culture' - the average /r/lewronggeneration person presumably does so out of some sort of rebellion against their perception of modern popular culture in 2020, and what's present and what's absent in it.

It's also, I suspect, related to the sheer cultural power the children born in the post-World War II boom - the baby boomers - have had in society. There were simply, a shit-ton of baby boomers, demographically. This lead to a situation where popular music could reflect the cultural concerns of one particular generation and still be massively popular. Rock music tends to be seen through a white male baby boomer perspective (and which thus focuses on the aspects of the music most important to those people). So it's relatively easy for young people to like a rock song or two, then learn about the music on wiki or google, and get something of that white male baby boomer perspective, and then identify the music of an older generation - the 1960s - as being the perfect counterpoint to modern American popular culture (and not least because baby boomers often think kind of thing about modern music themselves and are often relatively prominent culturally - Donald Trump, Bill Clinton, and George W. Bush were all born in 1946).

One thing that's different about such /r/lewronggeneration types today, compared to the people who grew up in the 1960s was that the baby boom was, more or less, the start of marketing to teenagers. Commercial interests were more than aware that a huge demographic was emerging (during a time of prosperity, too - baby boomers were therefore likely to have plenty of disposable income). It was a good demographic to advertise to, as a result. So you start to get radio stations, television shows, etc, explicitly aim to appeal to young people for what is, surprisingly, often the first time. Before this point, the idea of the 'teenager' was well established within the psychology literature - G. Stanley Hall, one of the more prominent early psychologists, had extensively argued for the existence of the teenager around the turn of the century. However, it was not really a way in which record companies thought about who their music would appeal to until the 1950s (which was more obviously broken down into regions and ethnicities). This is when you start to get 'rock'n'roll' as a marketing category designed to appeal to teenagers, with the rise of, for instance, Alan Freed's radio show in the early-to-mid-1950s. Freed played what he called 'rock'n'roll', which was basically a cross-section of the rhythm & blues of the era carefully curated to appeal to white teens, featuring the more upbeat and fluffy side of the genre. Big band swing was probably largely a young person's music in the 1930s - young people are probably more likely to want to dance, and it was a dance music! - but it wasn't really directly marketed in that way. Unlike the music of a 1960s, someone like Benny Goodman wasn't really seen as making 'the music of a generation' in quite the same way. This means that, in the 1960s, there weren't really deep wells of old music previously marketed to teens for a young person to discover.

The other thing to consider in terms of answering this question is simple availability. For old music to be available to consumers, two things need to happen - it needs to be made available by a record company to a record store in some way, and that record store has to want to sell it. In the 2000s, for the first time, the entire discographies of a band like the Beatles became available for the first time simply by pressing a button on a computer. Before the 2000s, someone who wanted to hear the Beatles likely had to purchase physical copies of their music, and whether a particular Beatles album was available in your local record store was the result of a record store putting out new releases of old product onto its shelves. This was most likely to be the record company printing new copies of the album - few record stores would keep stock that didn't sell. This severely limits the availability of old music, especially as the older generations who want copies of the music of their childhoods typically respond to different aspects of the music than they might have done as a teenager, often being less inclined towards noise.

How does this play out in the 1960s, in terms of the music of the 1900s? Basically, apart from all of the above, there's an added problem in that music recorded before about 1925 (and plenty of music recorded afterwards) was basically nigh-on-unlistenable to modern ears - and the ears of the 1960s too. It was only with the advent and wide availability of electrical recording in the 1920s that recording equipment began to capture a vaguely-realistic sonic and dynamic range. Recordings from the 1900s were basically almost unsaleable as far as record companies were concerned. So, no, young people in the 1960s were very unlikely to be listening to ragtime recordings made in the 1900s, as /u/randommusician points out.

However, if instead of ragtime, we focus on the music that's a little later - the 1920s and 1930s - do we see young people in the 1960s listening to music of this kind, despite all the caveats above? Well, yes. There's, broadly speaking, two different prominent cultures or subcultures involving young people which focused on reviving old music in the 1960s, and which weren't just an affection for the music of their parents.

The first would be the British movement which played what they considered 'trad jazz' (i.e., traditional jazz, which was to their minds jazz in the 1920s 'Dixieland' style). Of musicians who made music in that 1920s style in the 1960s, Acker Bilk was probably the most well-known - you can see a 1963 TV performance by Bilk and his band of 'Creole Jazz' here. Another group, in a way, who played in this style in the later 1960s was the much sillier Bonzo Dog Doo-Dah Band (who today are best known for including Neil Innes, who'd later work with Monty Python). The Bonzos covered a lot of 'trad jazz' tunes and in fact had a song called 'Jazz: Delicious Hot, Disgusting Cold' which was meant to be deliberately bad trad jazz.

Generally, the trad jazz movement was very popular in the late 1950s and early 1960s, and the movement for making blues music in the UK in the early 1960s - The Rolling Stones, etc - evolved out of the trad jazz movement (e.g., Alexis Korner who did a lot to grow that movement had played in Chris Barber's Jazz Band, one of the more popular British trad jazz groups of the 1950s). Speaking of Chris Barber's Jazz Band, another member of the band was Lonnie Donegan, who started the skiffle craze of the late 1950s in the UK (which was the origin of a lot of the 1960s British invasion groups - the Beatles famously started as a skiffle band called the Quarrymen) - skiffle being, essentially, sped up folk songs played with acoustic instruments (including things like 'washboard bass', where the bass was basically a string attached to a washboard, a bit like an overgrown bo diddley). The influence of Chris Barber on the British pop music of the 1960s is definitely a testament to how culturally important trad jazz was before Beatlemania and all that.