Henry's parents were Thomas Pettit (-1610) & Margaret Parker (-1615) - [m.1563 Ashdon]. They and the others I mentioned were all descendents of John Petyte (-1523) & Cecilie ? (-1546) of Ashdon, Essex.
Unfortunately, I don't know if they came over as part of the Huguenot refugees, or whether they were part of an older line derived from William the Conqueror's time. They may be related to the Cornish or Kent Pettits, but Ashdon, Essex is the earliest we've got to so far.
Do you have any info/dates for the Gilbert Petyt you mentioned?