1. What makes Parting Shot different or similar from Volt Switch, U-turn, and Memento? Are there any noteworthy advantages or applications that Parting Shot has over these other moves?
First of all, Parting Shot has to be handled differently from VoltTurn, as you can punish those two moves with Electric Immunity and Iron Barbs/Rocky Helmet, respectively. Instead, Defiant, Contrary and Taunt are the main issues. The advantages Parting Shot has are that the attack drops let you pivot in frailer mons, and it also has a niche use against Magic Bounce users as it can phaze them out (particularly helpful vs. Mega Sableye). However, the inability to deal damage means that you cannot wear down defensive checks with it, quite troubling for Pangoro (its only user) who cannot use its powerful 124 Attack with it. Parting Shot is better than Memento most of the time, as the ability to return to battle later on is more valuable than an additional stat stage drop (since switching cancels them out). PS: While there aren't any OU mons using this ability, Parting Shot is negated entirely by Soundproof, though I don't think this issue will ever come up.
2. Whether it's pivoting, offensive momentum, Memento-esque support, or something else, what is the "ideal" way to use Parting Shot? Why?
Because Parting Shot is a non-damaging move, it is probably better suited on a defensive mon as it does not require offensive stats to get more useful. Another good user would be a stallbreaker, which generally has more offense-oriented checks so that it can hit them with a stat drop, allowing another mon to come in more easily. Regarding speed, a Parting Shot user can actually be either fast or slow, where faster users would be able to weaken an attacker for a more defensive check and slower users can pivot just like with VoltTurn, with the added benefit of crippling incoming attackers. As a cherry on top, you can also screw over speed passers like Scolipede with a Parting Shot.