People still used both concurrently, at least at the time of Henry VIII. They had different usage.

Parchment was not phased out with the arrival of the printed press (which in its earliest form would use parchment, and makes books we call incunables). Official documents (such as charters) still used parchment for their longevity - even by the time of Elizabeth.

When it comes to literature and cheaper book production (thinking account books here), paper was already in use in the fifteenth century - often imported. You also had different quality of papers which were used for different purposes (or level of production: paper was cheaper than parchment, so clerics of lower level would have used it, but depending on quality, the paper might have been used by people of higher class who could afford but truly, all could use paper). If you are interested in the use of paper in the fifteenth century, I do recommend the work of Orietta Da Rold.

When it comes to correspondence, ie letter exchanges, paper would have preferably been used in the Tudor period.

Truly, both paper and parchment coexisted still in the beginning of the early modern period, the use of paper, the fact it was easier to obtain (with it being produced now on the UK by the early 16th century, on top of import) and cheaper to produce and afford (as it was made of reused cloth, rather than animal skin like parchment which requires a lot of preparation for a limited writing surface), became more mainstream. However paper was still not considered as sturdy as parchment, hence why official documents were still composed on parchment. Parchment was also being reused and recycled, at the time (you can scrap out what's written on it), and people still interracted with documents/books written on parchment (many tudor-period annotations have been made on medieval manuscripts). When it comes to the later Tudor period, paper was the medium used all around, especially with printing having become the main technology for carrying written texts, paper being widely available to and affordable by the greater public.

I'm a medievalist (PhD level) working on the manuscript culture of the late middle ages and on early print culture in England.

Edit: some sources about the early use of paper in England and in the early modern period.

• The Paper Trade in Early Modern Europe. Practices, Materials, Networks, eds Anna Reynolds and Daniel Bellingradt (Brill: Leiden and Boston, 2021) which includes an article by Orietta Da Rold: Networks of paper in late medieval England’, in The Paper Trade in Early Modern Europe. Practices, Materials, Networks, eds Anna Reynolds and Daniel Bellingradt (Brill: Leiden and Boston, 2021), pp. 148-166.

• Paper in Medieval England: From Pulp to Fictions (Cambridge University Press, 2020), ISBN: 978-1-108-84057-6, i-xx, 270 pp.