Depends on which Registry I can only speak for AMHR, but just looking at my AMHA rule book, I don't see this addressed directly, someone else may know that rule book better.
For AMHR - Page 237
Ribbons, braids, all decorative apppurtenances and stallion tack in halter conformation classes are not allowed unless otherwise specified in AMHR official class description. Bits are allowed on stallions.
Halter Division page 243:
Horses shall show with full manes and tails (no roach). Thinning, Pulling, shortening, inc. of a mane and tail is permissable. The mane and tail length shall be left to the decision of the exhibitor.,
Park Harness Horses page 254:
The horse is to give a brilliant performance with style, presence, finish, balance and cadence. Park horse to be neatly trimmed, well groomed, braided (same as USA Equestrian braid) with one braid in the foretop and one immediately back of the bridle path.
Jumper and Hunter page 257:
Hunters and Jumpers may show with braided manes and tails in the manner of their larger counterparts.