I've been studying the Cox Proportional Hazards model, and this question is glossed over in most texts.

Cox proposed fitting the coefficients of the Hazard function using a partial likelihood method, but why not just fit the coefficients of a parametric Survival function using the maximum likelihood method and a linear model?

In any cases where you have censored data, you could just find the area under the curve. For example, if your estimate is 380 with standard deviation of 80, and a sample is censored >300, then there is an 84% probability for that sample in the likelihood calculation assuming normal error.