I appreciate your information. It seems reasonable that Theresia could have taken the Perschy name either by adoption or because Janos Perschy was her biological father.
The source I used to establish my genealogy is LDS tape 700719 which I understand is from the Sankt Martin church record that was made for the diocese. Theresia Zotter’s birth record, entry 64 in 1863, does not include the name of her father, just her mother, Maria Zotter. But the Sankt Martin parish record for that birth might have the name of the father, perhaps added later.
I understand that the record made for the diocese every year was sent to Szentgotthárd, whereas the parish record stayed at the parish where it could have remarks added later.
When Theresia married, on 21 November 1893, using the name Theresia Perschy, her parents’ names listed in the Sankt Martin church record were Janos Perschy and Maria Zotter. So I know Janos Perschy was Theresia’s father at that time. What I don’t know is if he was Theresia Perschy’s biological father or her father by adoption.
That difference is only important to people like me trying to build a genealogy.
I understand the Catholic Church in Burgenland is in the process of making images of the old parish records. So I will wait for those images to become available and perhaps I will learn more.
Thank you for answering my question and the suggestion to ask at the the diocesan-archive. Best wishes.