I imagine some of you will not like this, so I apologize in advance and I humbly offer the following with all due respect.
I can’t think of many good reasons to change one’s mesorah, but shortchanging that mesorah just to sound like Zionists is (at least close to) avoda zara and, besides, one of the dumbest ideas I have ever heard, despite that many people do so (unfortunately, due to having been misled by the avoda zara of Zionism – see Rav Elchonon HY”D and many others).
Presumably, your real mesorah is to differentiate between taf and saf. Don’t let anybody change that unless a (non-MO) LOR tells you to do so.
(I specified “non-MO” because, since Zionism is an essential tenet of the MO faith, any issue that deals with Zionism must, by definition, be handled by a non-MO Rabbi.)
If you need to make your parents happy, I would suggest a possible compromise, however, that when you speak “modern hebrew” (as in mundane conversation), as opposed to, lihavdil, lashon haKodesh (for tefillah and learning), meaning for non-divrei kodesh, you can pronounce the saf like a taf. After all, that is how “modern hebrew” is spoken and if you’re speaking that language then perhaps it is okay, at least bishas haDichak, like in your case, to speak “modern hebrew” as it is “natively” spoken (ignoring the shmad and other issues).
I would also switch back to lashon haKodesh, even during mundane conversation, when referencing any davar sheBiKedusha, but I would ask a (non-MO) LOR.
But CH”V should your davening be sullied by “modern hebrew”. Our siddur (i.e. other than the Zionist insertions) is written in lashon haKodesh, not, lihavdil, “modern hebrew”. Your davening deserves no less than your best lashon haKodesh al pi your real mesorah.