Park Ball can't be bred down because it's not an obtainable ball type in any game that keeps track of the caught ball type. GS doesn't, HGSS replaces its role in the Bug-Catching Contest with Sport Balls, and while Pal Park uses Park Balls, they are not considered an actual ball type and when one of them is used, it steps aside and allows the previously obtained ball type to remain in the data. Likewise for other types of balls in the lore that are not functionally so, like the GS Ball or the Pester Ball from Pokemon Snap.
As far as implemented game logic goes, Master Ball and Cherish Ball are the only types that have special handling in the code: if either of those ball types would be passed down, a regular Poke Ball gets passed down instead.
