When Parallax and your CRM communicate about Dates, it's important to know which field is being referred to and how they work differently.
Target Start Date
Target Start Date (TSD) is a custom field that is typically set up during your initial connection to your CRM. Target Start Date allows Parallax to differentiate between when the Deal is being Sold, and when the Project should actually start, in case the Project won't start immediately after being Sold.
Target Start Date is listened to by Parallax, so as the Date is changed in your CRM, you can see the Project Start Date change, as well.
Close Date represents the date the Deal is being Sold within your CRM. Unlike TSD, the Close Date is only used as a back-up to set the Project Start Date if the Target Start Date is not selected at the time the Deal is pulled into Parallax (usually at creation). Once the Close Date is used to set the Project Start Date, Parallax will not listen to any further changes that happen to the close date.
It may sometimes be that a Target Start Date is not chosen at the time the Deal is brought into Parallax, in which case the Target Start Date can always be added later. Unfortunately, not all CRM's are compatible with Target Start Date, so it's also good to check Field Mapping for the CRM you're using to see if Target Start Date is one of the available fields - you can do that here.