  We study parabolic equations governed by integro-differential operators with
nonlocal components in some directions and local components in the remaining
directions. The setting contains the purely nonlocal, as well as the purely
local case. Our approach is based on an energy method allowing for jumping
measures that are singular or supported on cusps. In addition, the jumping
measure may depend on the direction. The emphasis of our study is on the weak
Harnack inequality and H\"older regularity estimates for solutions of such
equations. The main regularity estimates are robust in the sense that the
constants can be chosen independently of the order of differentiability of the
operators.
