Maybe I’m misunderstanding. I grew up in NYC, and “father” absolutely does rhyme with “bother”. Just listen to Run DMC: “they even bother my poor father cause he’s down with me.”
Looks like the difference is between the rounded and unrounded back open vowels /ɑ/ and /ɒ/. This site has an IPA chart where you can hear the differences. The father-bother merger hasn't happened in my (NE) accent, but I didn't know that pretty much everywhere else merged the two. Interesting that cot-caught merged for NE but not father-bother.
Younger New Yorkers do have the father-bother merger, but older New Yorkers don’t.
Also, Run DMC probably speak African-American English, which, as this map says, is generally independent of other dialects and not included on this map.
The Run DMC lyric actually sounds like the (previously unmerged) father vowel /ɑ/ went toward the bother vowel /ɒ/ than the other way around. I might even put it as /ɔ/ or /o/ when listening to the sounds on the IPA chart.
Whereas if you listen to the pronunciations on Merriam Webster father and bother it actually lists them both as /ä/, which is apparently a near-back vowel instead of back. I don't know which one NY does though.