Shared parameter joint models provide a framework under which a longitudinal response and a time to event can be modelled simultaneously. A common assumption in shared parameter joint models has been to assume that the longitudinal response is normally distributed. In this paper, we instead propose a joint model that incorporates a two-part ‘hurdle’ model for the longitudinal response, motivated in part by longitudinal response data that is subject to a detection limit. The first part of the hurdle model estimates the probability that the longitudinal response is observed above the detection limit, whilst the second part of the hurdle model estimates the mean of the response conditional on having exceeded the detection limit. The time-to-event outcome is modelled using a parametric proportional hazards model, assuming a Weibull baseline hazard. We propose a novel association structure whereby the current hazard of the event is assumed to be associated with the current combined (expected) outcome from the two parts of the hurdle model. We estimate our joint model under a Bayesian framework and provide code for fitting the model using the Bayesian software Stan. We use our model to estimate the association between HIV RNA viral load, which is subject to a lower detection limit, and the hazard of stopping or modifying treatment in patients with HIV initiating antiretroviral therapy