Quantile regression (QR) offers a flexible way to assess the effects of covariates on the quantiles of the conditional distribution of a random variable, given covariates. Since the effects of covariates can be assessed at any quantile, QR provides a better understanding of the effects of covariates comparing with traditional regression models. In this study, we consider a parametric conditional QR model for survival data with time-fixed covariates, and introduce a multi-stage estimation procedure to estimate the effects of covariates on the quantiles of marginal distributions of sequentially observed bivariate survival times. We model the dependency between survival times with a Clayton copula. Our estimation method is based on the martingale estimating equations. We discuss asymptotic and finite sample properties of the estimators obtained from this procedure. Finally, the method is illustrated by analyzing a colon cancer dataset.