What does lefty parents mean?
It means "left-leaning political" or liberal. These are parents who don't try to control everything their children do, like conservative parents, but tell them to think for themselves (use their noodle). Noodle is another word for brain, although I have no idea why.
To me lefty would mean left wing (in politics) or Communist.
Doctor P: read https
read https://www.betterhealth.vic.gov.au/health/healthyliving/left-handedness https://www.rd.com/health/wellness/left-handed-facts/
Most likely meaning is the parents are politically left or somewhat progressive as opposed to right or conservative.
I would understand it to mean that the parents are extremely liberal and bordering on socialist/communist. Not just regular liberals, but extreme liberals. "lefty" is a negative thing in US politics. The speaker might not actually mean that his parents are truly that radical, but he wants everyone to think it anyway.
Lefty means 'left handed'; it might also be a little dig, as 'off the wall', 'not usual'. It might also mean 'liberal' as in, 'socialistic leaning' or hippy. I'm sure they have your best interests at hear. "Use your noodle," as in, "Think for yourself and use common sense and you'll get by."