Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution. Developing efficient and accurate solution strategies that account for errors on the space, time and parameter domains simultaneously is a challenging task. Indeed, it is well known that standard polynomial-based approximations on the parameter domain can incur errors that grow in time. In this work, we focus on advection diffusion problems with parameter-dependent wind fields. A novel adaptive solution strategy is proposed that combines stochastic collocation on the parameter domain with off-the-shelf adaptive time stepping algorithms with local error control. The error estimation strategy is used to monitor errors relating to the time stepping and construct an adaptive approximation in which the interpolation error is controlled to a related tolerance. The efficiency of the error estimation strategy will be demonstrated numerically. The adaptive approximation is able to control the error by refining the polynomial approximation space through time and sequentially constructing a tailored approximation capturing the evolving uncertainty in the solution.