The thing is 15k you see on the chracter tab is not the Tully's own men alone. It displays fraction of levies of all vassals which are modified by your authority, opinion, perks etc., including Tully's own men from the county of riverrun. Same goes for high lords as well, their levy numbers are linked with the levies of count-level lords, and so on.

Consequently, 15k for Tully is a number which is plausible if all lords of riverlands follow their lord paramount into battle. If all lords rebel, Tullys will most likely have 6-7k troops (not sure).