If a convert's parents convert to Judaism, do they become his halachic parents? I understand that a ger is thought to be "reborn" upon conversion to Judaism; therefore, his birth parents are not considered his halachic parents. However, if parents convert to Judaism after their child's conversion, could they then become his halachic parents?
The Shulchan Aruch rules in YD 269:1 that a convert is permitted to marry his mother (who has since also converted) according to Biblical law, but it is prohibited rabbinically. Additionally, in 269:5 he rules the same way regarding marrying a female convert and her daughter. I'm not sure exactly what your test is for 'halachik parentage', but certainly as regards these arayot they don't seem to be related even after both have converted.
Rav Mari bar Rachel's was the son of Jewish woman -- Rachel was Shmuel's daughter and had been kidnapped and raped by non-Jews -- and his biological father was Issur Giyora, who converted after he impregnated Rachel. Talmud Bavli, Shabbat 154a; Bava Batra 149a; Yevamot 45b. Because his father converted later, he is identified by his mother's name and not by his father's.