This article targets the estimation of a time-dependent association measure for bivariate failure times, the conditional cause-specific hazards ratio (CCSHR), which is a generalization of the conditional hazards ratio (CHR) to accommodate competing risks data. We model the CCSHR as a parametric regression function of time and event causes and leave all other aspects of the joint distribution of the failure times unspecified. We develop a pseudo-likelihood estimation procedure for model fitting and inference and establish the asymptotic properties of the estimators. We assess the finite-sample properties of the proposed estimators against the estimators obtained from a moment-based estimating equation approach. Data from the Cache County study on dementia are used to illustrate the proposed methodology.