Well first, 'paper' doesn't mean any writing material. Paper is a specific material made out of cellulose fibers. (whether from wood, linen, cotton or hemp and other sources). Parchment is made from animal skins prepared into a thin scraped sheet (whether from goat, ox or other animal). Then there's for instance papyrus which is made from thin strips of papyrus pressed together. So paper is not parchment is not papyrus; they're different materials. (there's also vellum, which is a form of parchment so I'll skip that and just talk parchment. I'm also talking here about how historians use the terms; 'parchment paper' seems to be a marketing term these days but as far as we're concerned here those are two different things)

Parchment is more durable than paper, paper more so than papyrus, which is brittle and doesn't survive well in humid environments and didn't grow in much of Europe anyway. It was always far more expensive to produce parchment. First the raw material - animal hides - was more expensive, but it was also much more labor-intensive to produce, with stretching, hair removal, scraping it to the right thickness, etc. In the Middle Ages, parchment was used for things intended to be permanent; psalm books, or illuminated manuscripts, or important treaties. Purely temporary notes were written on reusable wax tablets. Paper occupied a status in-between.

During the Middle Ages, the art of paper-making spread in Europe, methods of producing it became increasingly sophisticated, and the quality improved. Thus parchment was being out-competed by paper by the end of the Middle Ages; parchment was still a better and higher-status material but good paper was just so much cheaper. As you say, parchment could be re-used (a palimpest) by washing and scraping, but so could paper, by methods that are in principle the same as how it's recycled today; by boiling it and freeing up the fibers which are washed and then added in with the fibers used when making new paper. Paper is really more easily reusable than parchment. (and had often been made out of recycled rags in the first place)

When printing comes around in the mid-1400s, paper quickly proves itself to be a better material to print on. Only a very limited number of things early on were printed on parchment (i.e. some Gutenberg Bibles). Printing on parchment would never have been an option anyway, though - there simply was not a big enough supply of parchment to produce any significant print runs on it. Not to mention, paper-makers quickly adapted to produce papers that were made specifically to be good to print on. The fact that they tried with parchment is a bit typical of incunabula though; which is a term for the earliest printed works (defined as pre-1500). The reason those books have their own term is that much more work was put into them than later printed books, because they tried to imitate hand-written manuscripts. The main text body was printed but ornamentation, titles, decorative initials and flourishes and even illumination were added in by hand afterwards. By 1500 or so, legendary printers like Aldus Manutius of Venice had developed typesetting into an art of its own. Some of the books they printed were almost modern in layout. (note that paragraph breaks weren't really a thing yet, though)

The production of incunabula is an indicator that for a time at least, handwriting on parchment remained the ideal. But with printing spreading and more and more books being available in print, it became increasingly difficult to justify funding immense effort in producing a single hand-copied volume when the same money could be used to buy a dozen printed works. (not an exact number) One who tried, though, was the abbot and cryptographer Johannes Trithemius, who directed his monks in De laude scriptorum ('In praise of scribes') in 1492:

Brothers, nobody should say or think: "What is the sense of bothering with copying by hand when the art of printing has brought to light so many important books; a huge library can be acquired inexpensively." I tell you, the man who says this tries only to conceal his own laziness.

All of you know the difference between a manuscript and a printed book. The word written on parchment will last a thousand years. The printed word is on paper. How long will it last? The most you can expect a book of paper to survive is two hundred years. Yet, there are many who think they can entrust their works to paper. Only time will tell.

In a historic irony, De laude scriptorum does not survive in any handwritten manuscript; we only know its contents today because of copies from when it was printed in 1494! It seems Trithemius' own work proved him wrong though about whether longevity was key to preservation; it seems quantity outweighed quality.

The development didn't stop for Trithemius or anyone though; so printing certainly played a part in killing parchment manuscripts, but skilled handwriting on parchment remained the higher-prestige option for centuries after. The advent of printing - but also the advent of more, better and cheaper paper - was significant to the decline of the handwritten parchment book. But not everything was printed or needed to be had in hundreds of copies. So parchment remained used for unique documents, papal bulls, treaties, other important legal documents; contexts where printing wasn't used but prestige and quality were important. For instance the US Declaration of Independence and Constitution were handwritten on parchment. To a limited extent there are still calligraphers using parchment to this day. Parchment and vellum also remained used in bookbinding, and a lot of that was reused; a not-insignificant amount of medieval texts have been found in the bindings of much later volumes. (picture)

So in summary; printing did affect the use of parchment but printing on parchment had little to do with it, as that was basically limited to some early experimentation. When it comes to recycling, paper was in fact cheaper and more reusable than parchment. But it was still quite expensive until the advent of paper making from wood pulp in the 19th century made paper became truly cheap and abundant. There was a real problem with people stealing book and documents to sell to merchants and manufacturers who needed waste paper to use as wrapping paper and for other purposes. For instance in 1801 they arrested a grocer in Södermalm here in Stockholm who'd been wrapping goods in important government documents; they found 37 volumes of stolen documents from the Svea Court of Appeals at the shop!

(Trithemius quote from Evylyn B Tribble, Ann Trubek (eds.) Writing Material: Readings from Plato to the Digital Age , Longman, New York, 2003)