May one bathe for pleasure purposes during the 12 months of mourning for a parent [i.e. Jacuzzi/spa]?

Yes. After the conclusion of Shloshim, starting from the morning of the 30^th day, the mourner may bathe and shower without restriction, and hence there is no restriction against using a Jacuzzi or spa. This allowance applies even for purposes of pleasure, and is unlike the restriction against different forms of entertainment which mainly applies being that it is done together with friends

See Rama 381:1 who only extends the custom for Shloshim.