Hey man, long time lurker here but Santa Monica resident.

LAX to Santa Monica is not too far but traffic is the rub. I would say it should be anywhere from 30 min to 45 min to get from LAX to Santa Monica Expo line. There's no parking at the line per se but you could park in one of the parking structures around 3rd street promenade. Generally, I don't think they're that expensive. I'd look up parking at 3d street premenade parking to see. Hope that helps!

Additionally, it might actually be easier to drive to a station further east on the expo line like Bergamont Station where there's less traffic than downtown santa monica, less people using the station and more likely able to find street parking near the line. I'd def look into that first.