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Skills penalty can be dealt in two ways:
Show Jumping and Western Pleasure competitions can distribute randomly a penalty to participating horses. This can be disabled with a à la carte VIP perk for players who are competitive in either discipline.
The other way is with some special coats such as the evil witch (from the Magic Hat customization BM object) of the bat (from the Bewitched Pumpkin). As far as I know, these cannot be avoided.