I just wanted to post some updates to my own research for Joshua's parents.
First, I went to Belmont county, OH, and the Joseph Parrish who died there left a will mentioning his sons Joseph and James. Additionally he mentioned his grandson Joseph, and granddaughter Charity. Based on the names, I believe this is the Joseph Parrish who married Charity Bosley. As it stands currently, there is no proof this Joseph is the same as Joseph Parrish of Shirley Twp, Huntingdon Co, PA, and there is nothing linking him to Joshua.
Second, I have new tax records that have Joshua in Frankstown, Huntingdon Co, PA in 1798, 1799, and 1800. He's listed as "removed" in 1800, which is also the first year he shows up in the Greenfield, Bedford, PA tax records. At this point his movements from 1793 until his death in 1840 are pretty well documented.
Finally, I've tried to find out more about John Parrish of Huntingdon Co, PA (in Allegheny Twp in 1800, in Springfield in 1810), however there is little to nothing available. Although he is in Huntingdon County in both the 1800 and 1810 census records, he is nowhere to be found in the tax records. This likely means that he was extremely poor and not only did not own any land of his own, he also didn't own a cow or a horse.
Personally, I think the connection between John Parrish and Joseph Parrish & Abarilla Parrish McGinnity (the latter two of Indiana Co, PA) is fairly well established, even though Abarilla's exact connection to John isn't known. And the circumstantial evidence of the baptism of Margaret McGinnity (Abarilla's daughter) in 1823 by Prince Gallitzin at the age of 16, with her godfather being John Parrish (not the same as above, this one is presumed to be Joshua's son John, born circa 1794), Joshua appears to be connected to them as well.
Basically to sum up, this mystery has not been solved and the award is still available. It is still my current belief that Joshua was indeed born in the US. Possibly he moved east after his father (?) Joseph Parrish of Shirley twp died and then moved back out west when Prince Gallitzin decided to start a Catholic church out there.