Just do a name change, and who would really know? Unless a person makes an effort to differ in appearance to be separate (as directed/commanded in Leviticus), or they keep a traditional Jewish name, I don't think most people would know or care one way of the other or would not really care.

It is the Jewish insistence on remaining separate and looking different to distinguish themselves as "god's chosen people", that makes them stand out as targets for discrimination and/or persecution. Yeah, I know that it isn't right, but that is just how things are and how humans generally behave and react to those who stand out.

Now they will say your protestant instead of catholic to avoid the embarrassment.

My parents gave me the same speech

There is an unfortunate bit (more than a bit) of truth in their statement. No matter how you feel toward any and all religion, there are bigots who have latched on to anti Semitic sentiment. And many Jewish people aren't even of Semite descent. Although I never experienced it, my father's side of the family were probably Irish Catholic as he told me stories from his Catholic uncles who experienced prejudice in the U.S. My father was born in 1919, and although he never experienced it, he used those stories to teach us about being against prejudice. At 10, in 66, I witnessed him give hell to a bigot. He also served in WWII, and was one of the first boot camp sergeants in an integrated military.
Anyway, there are a lot of bigots here and around the world. In 1972, after a lawsuit from a blonde, blue eyed woman, Louisiana changed their drivers licenses which would list anyone one two-hundredth black as negro. Boy did I go on for a long time. Simply put, yes, there are bigots who will center on you no matter your beliefs because of your heritage. (Note our current administration and its attacks on Hispanic peoples)