If you like to save 6€ in Neuschwanstein, just park along the road (it's allowed outside of the village, and plenty do) if you don't mind a few minutes extra walking. They've been increasing the parking fee every second year recently..
Regarding your trip to Lauterbrunnen (but generally the mountains), I'd prepared to have different plans of what to do depending on the weather. March might be sunny and fairly warm, with days getting longer and might be easily above 10C - or it could be the opposite, late winter storm, cold and freezing like hell, especially a bit higher up. So, keeping this in mind, I'd have some alternative ideas to do in case the weather plays bad..