The process of producing a copy of a text in large, bold print onto parchment is called engrossing. This term is still used today for the production of the final copies of laws. The usage of parchment as a preferred option to paper goes to the advent of print itself, as parchment was seen as a more formal, luxurious medium of writing, especially in the context of preservation and archiving. English laws were generally engrossed onto parchment in order to be suitably readable, durable, and formal. This was replicated for the production of the engrossed copy of the Declaration by Timothy Matlack and the Constitution by Jacob Shallus. It was only these versions which were actually signed. The public would have seen the Declaration only through paper-printed versions like the 200 Dunlap copies or innumerable copies in newspapers.

The top answer here is good, but too add a reference if you're interested in learning about the printing revolution check out Print, Manuscript and the Search for Order, 1450-1830 by David McKitterick. He explicitly goes over the transition from parchment to standard print. Parchment was preferred by wealthier individuals and for formal uses. At least in part because many were concerned with the longevity of paper. It's a decent read.