I frequently do the same thing. If I find a birth record with parents names, my first estimate is that each parent must have been born between 20 and 40 years before the birth of their first child, which gives me a period in which to hunt for the parents records.
On HANNA BRESIE, I have the following info:
Born bef July 29, 1770, Claverack, Columbia, NY. (Source LDS IGI File Extraction)
Baptised July 29, 1770, #1199, Reformed Church, Linlithgo, Columbia, NY. (same)
Parents: Johannes Bresie (same) and Maria Bein (notes below).
This Johannes Bresie has also been found with the following spellings: Jans D. Brazie, Johannes Bresee, Johanis Brisie, Jans D. Brissee.
This Maria Bein has also been found with the following spellings: Marya Ben, Marya Ben, Marytje Ben, Maria Benn.
I'll answer your last question before proceding. My mother's family always spelled their family name "Brasie". In tracing back five generations I encountered spellings of Brazee and Brazie. When unable to connect William Brazie Jr, born abt 1817, I began researching all NY Brazies, and found over 50 different surname spellings. When I find someone to add to my file, I document the name as originally found, then assign that person an alias with the surname "Brazie", which is the form I use for sorting family groups and researching. When I use "Brazie", I probably found the original record containing a different spelling. Besides, there is a Brazie entry in Genforum, but not a Brasie group. Now back to Hanna's family.
Bear in mind that I don't claim my file to be error free, but I do try to confirm entries with a reliable source before entry into my file. That being said, here is how I have configured Hanna's family:
Johannes Brazie b 1730-1750
+ Maria Bein b Bef. June 25, 1752
-Hanna Brazie b Bef. July 29, 1770, Claverack, Columbia, NY
-Teunis Brazie b June 23, 1773, Manorton, Columbia, NY, m Eva Snyder
-Catharina Brazie b November 20, 1779, Manorton, Columbia, NY
-John J Brazie b September 05, 1784, West Copake, Columbia, NY, m Lavina Konklin
-Hugh Brazie b October 12, 1788, West Copake, Columbia, NY, m Caty Concklin
-Dorothea Brazie b November 29, 1796, West Copake, Columbia, NY
I don't have Johannes parents yet. I feel sure his father is in my file, but havn't made the connection yet.
On Catrina Bresee, I have lots in file by names Catharina, Catherina, Catherine, Cathrina, Cathrine, Catlinche, Catlyntje, etc, but can not find a connection.
Back to names and spellings for a moment. I contend that my Brasie line descends from Christoffel Brussi, b abt 1653, Netherlands, Belgium or France, and Christina Claesgen, married in the Albany NY area abt 1678. In those days, the population included people of Dutch, German, French, Indian, and other heritages. Few were literate. Often, names were written down from the sounds (with dialects) pronounced by others. Occasionally, "Pressie" or a similar variation shows up in the records of families that I consider part of my line, but it was not very comon. When I do find it, I look for a connection, but this form of spelling is not primary for me. There are a few other lines, including Bries, Breesee, Breesse, Bracy and Bracie which, except for the occasional accident of spelling, I consider separate and distinct lines from my Brasie line.