So what happens are there are a certain number of permit parking spaces. Most of them are for people who buy monthly passes, but there are other people who can park there as well, such as the long term parking permits, single day permits, etc. Before 10 AM, you must have a permit in order to park in one of these spaces. Say there are 100 permit parking spots. They sell 70 as monthly permits, then hold the other 30 for other uses. Something like that. That's how they guarantee that you will have parking prior to 10 AM.
After 10 AM, any unused permit parking becomes "fee" parking, which is basically first come first serve at that point. You park there, you pay the machine, even if you don't have a permit.
If you have a long term parking permit and you show up before 10 AM, BART has (hopefully) only sold enough permits to fill the permit parking but not any more than that, so you are guaranteed a parking space. If you show up after 10 AM, then you are still allowed to park in the permit parking or the regular parking. The problem is that for popular stations, there will be literally no parking spaces during the weekdays after 10 AM. If I'm going to San Francisco on a weekday after 10 AM, I don't even bother trying to park at a BART station anymore (at least the ones where I live), I just drive there. So you will have your permit, but no parking spot. After 10 AM, there is no mechanism for BART to hold a parking spot for you. I do not recommend using one of these long-term parking permits if you plan to show up after 10 AM on a weekday, unless you know your particular station well enough that you know there will be parking after 10 AM.
Weekends are no problem finding parking generally, at least the ones that I have been to.