The truth is that you never really know if your data follows an exponential, Weibull, etc. (with the exception of data you simulated yourself). In fact, it's a very safe assumption that your real world data does not exactly follow any parametric distribution. As such, you are picking a model which best resembles the patterns you typically see in that type of data and assume that this is a good approximation of the actual distribution.
A fairly simple method for assessing whether the parametric model you assumed is appropriate is to compare to the Kaplan Meier curves themselves with the fitted model; i.e. plot the fitted KM survival curve and the fitted parametric curve on the same plot. If the parametric model is appropriate, then the KM survival curve should basically resemble parametric curve, but with some random noise. If it is a bad fit, you may see some systematic differences between the two curves.
That being said, in most cases I would greatly prefer the KM curves for estimation, mostly due to them being more robust and basically as efficient (the survival estimates for which using a parametric model has much lower standard error than KM curves are the estimates which rely heavily on the parametric assumption, so the gain you get is from making assumptions for which you cannot properly assess the appropriateness). However, there are some cases in which you may want a more complex model and so you must use parametric models, such as cure-rate models.