The last time I checked the Park and Fly rate at the LAX Courtyard, I thought the rate was pretty steep (over $200 per night).
Here's an idea. I did this a few years ago.
I've got a Marriott Premier credit card. This means that I get an e-certificate for one free night at a lower category Marriott each year (in the first year of card membership, I think it's a category 1-4, and after that it is cat 1-5).
If you do a search on LAX at Marriott.com, you will find that there are several hotels not too far from LAX. So, I called a few of them and asked if it would be possible for me to leave my car parked out front (at my own risk) if I were to spend one night there using my e-certificate. I think it was the Springhill Suites LAX/Manhattan Beach (category 3) that agreed. This hotel does not have shuttle service to and from LAX, but they are only a few miles from LAX.
So my wife and I parked our car out front, checked in, spent a few minutes freshening up in our room, and then asked the front desk to call us a taxi. It was about $20 to the airport. We were away for a little over 2 weeks and then took another $20 cab ride back to the hotel to pick up our car. Saved us quite a bit compared to the Park and Fly rate at the LAX Courtyard.
The other alternative for us is just to use a limo service. From where I live, the cost of a private limo to and from LAX is not all that much greater than the Courtyard's Park and Fly rate.