Among my many home-builds and individual clients, there are eight pata drives working fine in sata ports with adapters.
You should not have a problem. BUT, there are caveats.
Buy the adapter only from Newegg or Amazon.
Make sure the adapter takes you from pata-to-sata.
Most of these adapters are NOT bi-directional; most of them take a sata drive to an IDE connection only (just the one way). Be careful.