This paper discusses the stability analysis of linear parameter varying systems with a parameter-dependent delay where the parameters are assumed to be stochastic piecewise constants under spontaneous Poissonian jumps. Based on stochastic Lyapunov-Krasovskii functionals, we also provide sufficient synthesis conditions for the gain-scheduled state-feedback controller with memory in terms of parameter-dependent linear matrix inequalities (LMIs). Such synthesis conditions are computationally intractable due to the presence of integral terms. However, we show that these LMIs can be equivalently represented by integral-free LMIs, which are computationally tractable. Finally, we illustrate the applicability of the results through examples.