This kind of navigation is not yet available. It is only possible to change the date using a full or partial PostBack (having DayPilot Pro inside an UpdatePanel from ASP.NET AJAX library).
However, I plan to add date navigation API in
DayPilot Pro 3.8. It should include integration with System.Web.UI.WebControls.Calendar (date change using a partial CallBack) and a client-side API for moving the date. Then it will be possible to have custom buttons that trigger StartDate change.