I am adding information about possible relatives here. These are people I have come across in my research who *may* be closely related to Joshua, however the "proof" is limited and circumstantial and quite frankly confounding.
Here are some of the possible relatives I have come across. It is highly uncertain whether any of these are actually close relatives.
The first is Joseph Parrish of Shirley Twp, Huntingdon County, PA. Joseph lived in Shirley Twp from at least 1783 to 1789. He does not however appear in the 1790 census, and he also disappears from the tax rolls at this point in time. Joseph served in the militia of the area during the Revolutionary War. His history is unknown but it's possible he was the son of William and Susannah Parrish of Maryland. The reasoning is that there is a man named Merryman Price living in Shirley Twp with Joseph Parrish. The aforementioned William Parrish had a son, William Parrish who married a woman named Keturah Merryman. Prior to this marriage, Keturah Merryman had been married to a man named Thomas Price. Although a son named Merryman has not been found to date for them, circumstantial evidence says he has a good chance at being their son. If this is the case, it would make Joseph Parrish a sort of step-uncle to Merryman Price. If this is Joseph, son of William, he was married to a woman named Charity Bosley in 1768 in Maryland. They were kicked out of the Society of Friends for marrying. Additionally if it's them, at some point they moved on to Ohio, as their graves are in Belmont, and there is a likely grandson named Bosley Parrish.
Abarilla Parrish is another interesting subject. She was born about 1780 in Pennsylvania (?) and married a man named James McGinnity. They lived in Indiana County, just west of where Joshua was living with his family in Cambria County. Abarilla & James were Protestant, while Joshua was Catholic, but of course Joshua was a convert. Abarilla's daughter Elizabeth McGinnity married a man named Joseph Parrish who was supposedly "her uncle". If so it's a very close marriage!
Continuing along Abarilla's line, in the baptismal records for Cambria county, a Margaret McGinity born in 1806 (correct year for Abarilla's daughter Margaret) and was baptized in 1823. She's listed as the daughter of "Priscilla & Jacob McGinity". As the names were latinized, Jacob is of course the Latin form of James, and since Abarilla is an unusual name, the priest likely chose a form he thought best fit, hence "Priscilla". The real kicker is that her godfather is a John Parrish and the only John Parrish around at the time was the nearly 30 year old son of Joshua Parrish. The only drawback here is that Abarilla's daughter Margaret McGinnity married a Protestant, so it doesn't really make sense as to why she would be baptized Catholic at the age of 17.
Another potential relative is a John Parrish of Allegany, Huntingdon County, PA who shows up in the 1800 census. He's living in the same town as Prince Gallitzin (the priest Joshua followed out to Cambria county) and he has a boy living with him would be the right age to be the Joseph Parrish above, married to Elizabeth McGinnity. Likely he is in the 1810 census in Springfield Twp, Huntingdon County, PA as J. Parish where he still has a boy the right age to be Joseph and there's still a woman living with him, presumably his wife. If related, it is my assumption John Parrish is a brother of Joshua as I do not believe he would have been old enough to be his father.