Have you considered driving to the *next* train station then? I know it adds on more time, but if your home station doesn't have any parking, and you can't park in the streets nearby, then perhaps consider going to the next station?
How far away is the closest street from the train station, which doesn't have construction business going on?
Alternatively, maybe chat with the staff at the train station and see if they notice what time the car park seems to get full (if it's in full view of the station - if it's directly next to it etc), and plan to be there before hand. I know it's not exactly ideal to be there much earlier than you have to be, but if it avoids the stress of trying to find a park and copping a fine, it's worth the hassle.