My cleric was paralyzed by a yeti, then one shot critted to unconsciousness. If my comrade heals me, am I still under paralysis and required to make a Con saving throw? Or does the paralysis shake off after coming up to positive hit points?
The Paralyzed condition lasts until a minute passes; in general, conditions tell you when/how they end
The section on "Conditions" states:
[...] A condition lasts either until it is countered (the prone condition is countered by standing up, for example) or for a duration specified by the effect that imposed the condition.
In this particular case the Yeti's statblock states (I have assumed the Yeti's Chilling Gaze feature is what paralyzed you):
Chilling Gaze. [...] The target must succeed on a DC 13 Constitution saving throw against this magic or take 10 (3d6) cold damage and then be paralyzed for 1 minute, unless it is immune to cold damage.
Unless something explicitly says otherwise, the paralyzed condition that Chilling Gaze causes will not end until the 1 minute is up; one such explicit exception is the lesser restoration spell, which states:
[...] You touch a creature and can end either one disease or one condition afflicting it. The condition can be blinded, deafened, paralyzed, or poisoned.
Exhaustion (p. 291). The following sentence is appended to the last paragraph: “Also, being raised from the dead reduces a creature’s exhaustion level by 1.”
There would be no reason to reduce a creature's exhaustion level upon resurrection unless death did not reduce it to 0 already. Thus exhaustion levels persist even through death, let alone just unconsciousness, this would similarly apply to other conditions as well.
It is noteworthy that while unconscious you would continue to make saving throws to end the paralysis from Chilling Gaze (if you wanted). This could end the effect before the 1 minute is up, and is possible because ending the Paralysis requires a Constitution saving throw -- both the Unconscious and Paralyzed conditions only make you automatically fail Strength and Dexterity saving throws.