We consider an abstract formulation for a class of parabolic quasi-variational inequalities or quasi-linear PDEs, which are generated by subdifferentials of convex functions with various nonlocal constraints depending on the unknown functions. In this paper we specify a class of convex functions {φ(v; ·)} on a real Hilbert space H, with parameters 0 ≤ t ≤ T and v in a set of functions from [−δ0, T ], 0 < δ0 < ∞, into H, in order to formulate an evolution equation of the form u′(t) + ∂φ(u;u(t)) ∋ f(t), 0 < t < T, in H. Our objective is to discuss the existence question for the Cauchy problem.