I haven't lived with my parents for more than 6 months since I was 17 years old. I went to college at that age, and the only time I have been back home for a substantial time is a 6 month period when I moved to LA, couldn't find a job for 9 months, and eventually had to move back to NYC. I eventually moved back to LA after finding fulltime work, but for another year I was living with roomates in Monetery Park. I was 27 when I was finally able to live comfortably on my own. Between 17 and 27, I went to college, grad school, USMC, and moved west to Los Angeles. I now have a great job with a very competative salary. I may not have the savings that I would like, but I think i am doing better than most. However, the life experiences that I have gained over that time are irreplaceble. However, like someone mentioned earlier, my relationship with my parents (which were already strained) have been badly damaged since I have been away for so long.
As far as the OPs question, I think he should stay at home for 2 years and and save as much as he can. Having a solid $20k at 24 is a great way to start your life. You will need a car in LA, and since you are agetting in to the hollywood scene, I imagine that you will want to "floss". Please resist the urge to buy a car that you cannot afford. Get your self an older model civic or fusion for $10k and be done with it. If you think your car will get your laid, then your are in for a surprise. Most of the people in the industry will be dirt poor as well; no need to be another pretender. Work on your social skills and your game and it doesn't matter if you live at home, with roomates or in a pad in Malibu; the girls will come. With that said, you'd get a higher quality woman in your mid to later 20's than you are going to get now anyways.