I believe the following reference is Peter's birth record:
"England and Wales Non-Conformist Record Indexes (RG4-8), 1588-1977," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:VWZ7-KMH : 11 December 2014), Derrie Pety in entry for Pierre Pety, 22 Feb 1645, Baptism; citing p. 44, Spitalfields, London, record group RG4, Public Record Office, London.
The first name is Pierre, AKA "Peter", and the last name is phonetically the same. The date of birth is the same in London. This source lists parents as Derrie Pety and Marie Pety. This registry includes all Nonconformists, which includes Hugeonots, but in 1645 this would also include Catholics, Protestants, Jews, and Quakers.