No, there is no parking at all at Woodside train station. You might be able to score some weekend parking in the adjoining neighborhoods, but that's the standard city gamble.
I take the train out to Fire Island all the time and the Woodside train station is my access point as well. Generally, I'll take the Q18 to the Woodside train station and then grab the LIRR. On the way home, its almost always a Sunday, so Q18 service is marginal. I'll splurge for a cab ride from one of the black cars. If it is really nice and I feel into it, I'll hump it from the station to home. Its about a 2 plus mile walk, pretty safe and its not the end of the world to do one of those once in a while.
I think if you work out the costs for your situation, you'll still be saving money by doing something like that, in particular with the cost of gas. And as a plus, drinking and driving is no longer an issue.