If I use Fantina and then my Flying Pikachu V with Big Parasol attached uses its "Fly" attack and successfully lands heads, would my opponent's Giratina V's "Shred" attack do any damage to Flying Pikachu V or not?
Shred isn't affected by "any effects on the Defending Pokemon"; so the effects from Fantina and Big Parasol and Fly would all be ignored, and Shred would do damage.
Source: TPCi Rules Team (2022-09-29)
If my opponent uses Guzzlord's "Red Banquet" attack and my Active Pokemon with Big Parasol attached gets KO'd, does my opponent get to take an extra prize for "Red Banquet" or not?
Big Parasol prevents all effects of attacks from your opponent's Pokemon done to your Pokemon; but taking an extra prize is a game effect, not "an effect done to your Pokemon", so your opponent gets to take the extra prize.
Source: TPCi Rules Team (2021-04-08)