No, there is no rule stating that you must have a white saddle pad at competitions, but at dressage competitions it’s a common/an unwritten rule to use a white saddle pad. 

In eventing on the other hand, strong colors are very common. There are however no rules regarding which color should be used in competition. 

Equestrian Stockholm offers a wide range of saddle pad colors, from classic white for competitions to more colorful options.