Parametric multiple comparison procedures (MCPs), taking into account the correlations between test statistics, are expected to be more powerful than their counterpart non-parametric procedures, and can be a useful testing strategy for clinical trials with subgroups and interim analyses. This is because test statistics are inherently correlated between the interim and final analyses as well as between the subgroups and overall population. We derived a flexible parametric MCP based on an approximation of correlations between stratified log-rank test statistics and closure principle, referred to as Closed Testing Parametric Procedures (CTPPs). We compared the performance of CTPPs to other parametric and non-parametric MCPs. Simulation studies showed that the type I error rate of a CTPP was strongly controlled. Compared to weighted Bonferroni, group sequential Holm procedure, and Spiessens and Debois' method, the CTPP exhibited the greatest statistical power and the statistical power of the group sequential Holm procedure was only slightly lower.