Theunis Jansz. Pier was born circa 1600. Chris Brooks and I researched and found Theunis' baptism. His parents' names are now known!
We found a great deal on the family -- and our jointly authored article
Origins of the Pier Family in the Netherlands and an Update of Their Connection to the Ostrander Family by Lorine McGinnis Schulze and Chris Brooks, was published in the July 2000 issue of the New York Genealogical & Biographical Record.
Theunis married Jannettie Arentsdr. on September 14, 1624 in Amsterdam Netherlands. Theunis Jansz's sons Jan and Arent, were the immigrant ancestors of the PIER family in America. See
http://www.olivetreegenealogy.com/nn/surnames/pier.shtml for more