Competing risks arise with time‐to‐event data when individuals are at risk of more than one type of event and the occurrence of one event precludes the occurrence of all other events. A useful measure with competing risks is the cause‐specific cumulative incidence function (CIF), which gives the probability of experiencing a particular event as a function of follow‐up time, accounting for the fact that some individuals may have a competing event. When modelling the cause‐specific CIF, the most common model is a semi‐parametric proportional subhazards model. In this paper, we propose the use of flexible parametric survival models to directly model the cause‐specific CIF where the effect of follow‐up time is modelled using restricted cubic splines. The models provide smooth estimates of the cause‐specific CIF with the important advantage that the approach is easily extended to model time‐dependent effects. The models can be fitted using standard survival analysis tools by a combination of data expansion and introducing time‐dependent weights. Various link functions are available that allow modelling on different scales and have proportional subhazards, proportional odds and relative absolute risks as particular cases. We conduct a simulation study to evaluate how well the spline functions approximate subhazard functions with complex shapes. The methods are illustrated using data from the European Blood and Marrow Transplantation Registry showing excellent agreement between parametric estimates of the cause‐specific CIF and those obtained from a semi‐parametric model. We also fit models relaxing the proportional subhazards assumption using alternative link functions and/or including time‐dependent effects. Copyright © 2016 John Wiley & Sons, Ltd.