This article proposes a modeling strategy to infer the impact of a covariate on the dependence structure of right-censored clustered event time data. The joint survival function of the event times is modeled using a conditional copula whose parameter depends on a cluster-level covariate in a functional way. We use a local likelihood approach to estimate the form of the copula parameter and outline a generalized likelihood ratio-type test strategy to formally test its constancy. A bootstrap procedure is employed to obtain an approximate $p$-value for the test. The performance of the proposed estimation and testing methods is evaluated in simulations under different rates of right-censoring and for various parametric copula families, considering both parametrically and nonparametrically estimated margins. We apply the methods to data from the Diabetic Retinopathy Study to assess the impact of age at diabetes onset on the time to loss of visual acuity.