side horse vs gymnastic horse

side horse and gymnastic horse both are nouns.

Word	Noun	Adjective	Verb	Adverb
side horse	Yes	No	No	No
gymnastic horse	Yes	No	No	No

As nouns, gymnastic horse is a hypernym of side horse; that is, gymnastic horse is a word with a broader meaning than side horse:

side horse: a gymnastic horse with a cylindrical body covered with leather and two upright handles (pommels) near the center; held upright by two steel supports, one at each end
gymnastic horse: a padded gymnastic apparatus on legs

Other hypernyms of side horse include horse.

side horse (noun)	gymnastic horse (noun)
a gymnastic horse with a cylindrical body covered with leather and two upright handles (pommels) near the center; held upright by two steel supports, one at each end	a padded gymnastic apparatus on legs