I have now found more information about the Parkes family I am looking for, but still can't connect them back to the right family in England.
I am researching Joseph and Mary Parkes, whose first 4 children were born in England (Henrietta, 1826; Josiah, 1828; Sarah, 1829; and John, 1831). Mary and the children emigrated in 1833--Joseph came ahead, but I haven't found him yet.
With Mary on the ship was Sarah Parkes Beckley, wife of William Beckley. I assume that Sarah must have been Joseph's sister. The Beckleys also named their first daughter Henrietta--she was baptized 23 May 1825 in Brierley Hill (which is where all Joseph and Mary Parkes' children except Josiah were baptized).
Another Parkes who named the first daughter Henrietta was John (married to Jane), born about 1806 in Brierley Hill (according to the 1851 UK census).
I would love to contact anyone who has information on any of these families.