Obviously the music industry doesn't have a problem doing it still in other ways (see Britney Spears, etc), but we don't hear songs flat-out admitting that they'd like to get with a minor on the radio nowadays. Were the kinds of lyrics below a holdover from earlier days, before the invention of adolescence, when mid-to late-teens was a perfectly acceptable age to get married and have children?
For example:
The Beatles, 1963: "She was just seventeen, and you know what I mean"
The Zombies, 1968: "What's your name? Who's your daddy? Is he rich like me? Has he taken any time to show you what you need to live?"
Even Gary Puckett's "Young Girl," (also 1968) while admitting that sexualizing an underage girl is a problem, treats the issue in more matter-of-fact way. I doubt modern lyricists would even go there.
Keep in mind the people singing/writing the songs. "I Saw Her Standing There" was written in 1962, when Paul McCartney was 20 years old. Rod Argent would have been 22 when he he wrote "Time of the Season."
Compare that to Justin Bieber's "Baby": "When I was 13, I had my first love..." It's the same age difference as between McCartney and the object of his affections. (Bieber was 16 when the song was released.)
The average age of marriage has climbed quite a bit since the 50/60s. In 1950 the average age to get married for women was 20.3. There would have been a fairly large number of late teens getting married. It has since risen to 26.1.
With regards to the Beatles, there are two factors at work here. The age of consent in the UK is 16, so it's not exactly illegal. Secondly, "seventeen" is a much better fit to the lyrics achieving a match in terms of meter and alliteration.
Age of consent varies in the US from 16-18. Canada raised the age from 14 to 16 in 2008, but allows down to 12 if the age difference is less than two years.
There are a lot of songs through the 1950s going all the way into the 1980s all about young women who have reached some sort of age of marriageability, "womanhood" (whatever that means), or - perhaps most likely - consent. Most of these tunes are in rockabilly or rock-and-roll styles, which mostly came out of the American South, where the states had a median age of consent of 16. And once you get enough of those kinds of tunes, you get a lot of copycats, which is how the "sweet sixteen" genre of love songs becomes briefly ubiquitous.
So saying "she's only 16, you know what I mean" is in many of these contexts pretty much saying "She just became legal!" Not that she's marriageable - although that may very well be the case as well.
Many of these songs are also being sung from the perspective of a younger person, and in the context of a society with a much lower average age of marriage, and in a time when the concept of adolescence as we know it was still developing.
So, these songs can be viewed as more innocuous if they're coming from a) a singer close in age to the subject, b) a society where 16 is legal, and c) a culture that views older adolescents as adults.
What I would love to see - and what is, unfortunately, beyond my knowledge - is some information here about the concept of womanhood, since many of the grossest songs in this category ("Thank Heaven for Little Girls" and all of the sweet sixteen tunes) tend to work with this concept.
The Broadway musical Bye Bye Birdie plays with a lot of these ideas - age of consent, girlhood-to-womanhood, dumb rock songs, creepy dumb older guys, the invention of adolescence, and youth culture - pretty expertly, and while we know it today as a musical for high schoolers it's kind of an excellent satire that happens to have been written right in the middle of the era in question (1960). The content of the musical - which was very, very popular - suggests that even at the time a lot of people found the girlhood-to-womanhood point arbitrary, the sexualization of young girls by older men gross, and the songs dumb and goofy. The plot concerns a rock-and-roll star who takes part in a staged publicity kiss with a teenage fan, and the hijinks that ensue.
Finally, in the context of the song "Time of the Season," the "who's your daddy" almost certainly refers to a lover, not a father. (When's the last time somebody walked up to somebody they wanted to have sex with and said "Who's your daddy?" while meaning "what's your father's name"?) It can best be interpreted as "So, who are you banging right now?" A lot of other references to "daddy" and "old man" in other songs can be interpreted this way.
Source: "Age of Consent Laws [Table]," in Children and Youth in History, Item #24, http://chnm.gmu.edu/cyh/primary-sources/24 (accessed February 9, 2015). Annotated by Stephen Robertson