We develop flexible multiparameter regression (MPR) survival models for interval‐censored survival data arising in longitudinal prospective studies and longitudinal randomised controlled clinical trials. A multiparameter Weibull regression survival model, which is wholly parametric, and has nonproportional hazards, is the main focus of the article. We describe the basic model, develop the interval‐censored likelihood, and extend the model to include gamma frailty and a dispersion model. We evaluate the models by means of a simulation study and a detailed reanalysis of data from the Signal Tandmobiel study. The results demonstrate that the MPR model with frailty is computationally efficient and provides an excellent fit to the data.