Western Europe is not exactly known for its use of mounted archery. But, illustrations that accompany religious texts indicate Europeans were well aware of both mounted archery and composite bows (see video). Interestingly, one French bowhunting manual (the Book of King Modus) contains a riveting description of how to hunt deer and hares on horseback using a longbow. The archers are described as shooting some animals while in motion, and others while motionless, simply using the horse to efficiently move into a favorable position for a static ambush. There is even a recommendation for a technique that could be interpreted as a Parthian shot!
"When he preceives they are quiet, he may approach very cautiously, until in a favourable position for shooting; in other words, until he can discern the side of the fattest of the herd fully exposed to his arrow: let him then halt, and handle his weapons. Let him shoot with all his force, drawing the arrow fully up to its head, and levelling his bow-hand at the spot where he wishes to pierce the game."