I've read about dismounted knights participating in sieges, assaulting or defending fortifications, but I've never read about any other mounted force used in this way. How would cataphracts, horse archers, or other cavalry be deployed during a siege?

Ottoman mounted forces were also used during sieges and contributed in wide variety of roles. Ottoman army as a military force had up to half and sometimes even more of its troops as cavalrymen during its classical period. As the Ottoman empire was involved in numerous wars which also involved a large number of sieges, the cavalry forces were also naturally involved in most of these.

Ottoman cavalry at this time, besides various auxiliaries, was primarily divided between standing elite Kapukulu Sipahi and more numerous provincial Timariot Sipahi. Kapukulu cavalryman was better equipped, more heavily armored and served as heavy cavalry and in comparison provincial cavalryman was lightly armored and served as light cavalry. Both Kapukulu and Timariot cavalrymen had their primary role as horse archers but Kapukulu generally acted in reserve and were more likely to be involved in close quarter combat when they were fighting.

Timariot cavalrymen were deployed dismounted generally to provide ranged support during sieges or defensive operations. As they were lightly armored, they were most commonly placed behind infantry, war wagons, makeshift fortifications or natural obstacles. They assisted infantrymen with constant volleys of arrows. However some of the higher ranking and better armored Timariots would also be placed among the infantry to act in officer role and to provide examples of courage and valor to inspire the men. They were almost always involved in final assaults to provide crucial ranged support on walls or from behind the infantry in breaches. Kapukulu cavalrymen, fewer in numbers and made up of better trained and more experienced men were used in sieges usually as reinforcements, either to close up a siege in its final phases or only to provide assist in decisive assaults if needed.

As a side effect, because of similarity of siege warfare and naval warfare during the time period of roughly beginning of 15th century to end of 16th century, as well as Ottoman cavalry's choice of equipment, they not only participated in sieges but also in naval battles and landings in similar roles.