I did it eventually. I'll write the solution for my problem here maybe there will be somebody that will need it.
I said I can only retrieve the values of the parameters in the BeforeRender event of the CrystalReportViewer object. That's not true though. Once you post the page back to the server you can retrieve the collection of values from this.Page.Request.Form collection. Therefore I was able to retrieve the StartDate and EndDate pretty early in the lifecycle process of the page and now I am saving the data in the DB and the report shows the updated data.
This is how the code looks like:

DateTime startDate = DateTime.Parse(this.Page.Request.Form["StartDate0DiscreteValue"].ToString());

where StartDate0DiscreteValue is the name of the first control that the input parameters page renders.

Hope it might help somebody.

Radu