Consider using a range of dates filter. It uses a slider to select both the start and end dates, but if the user clicks on either the start or end date they can type in the date or use the calendar to select a date. The following Online Help article provides instructions for setting up a range of dates filter.
Parameter controls can be used to create start and end date filters that will be drop downs, but currently using one parameter to filter another parameter is an unsupported feature in Tableau Desktop. The following Knowledge Base article provides instructions for creating start and end date filters.
I hope this helps!
Once the user selects a start date...the end date drop down should have value equal to or greater than the start date.
This is a good and obvious functionality which I am afraid isn't featured in Tableau, not even in 9.0. Interestingly, I am not aware of anyone specifically requesting this in Ideas. The closest I can think of is Dynamic settings of settings. If you don't create an idea for this, I hope to remember to do so myself.
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