I think it's worth an update since this question was asked nearly 5 years ago. A combination of personal experience and local news coverage (example) informs this answer.
In a region where a lot of folks already like to get outside, the pandemic has greatly motivated even more people to do so. As a result, Paradise is an absolute zoo on summer weekends. Regardless of whether the parking lots are technically full or not, if you'd like to have at least some semblance of peace on your visit I would recommend arriving before 9am, well before if possible. By noon it's a madhouse.
If you really just need to see the sights and don't care about the congestion, you can probably find parking within walking distance of Paradise most times of day. It might be a very long time to find a spot and a long walk, though.
Sunrise doesn't fill up quite as early (maybe another hour to 90 minutes of leeway) but when it's full cars are held at the park entrance until spots become available. Sunrise also has less capacity, so if you're there when it's full the experience will be somewhat more pleasant than Paradise when it's full. But still not great.
Of course, you'll also want to be leaving the park early or very late to avoid getting stuck in line on the way out. Arriving rather late is also an option.
On holiday weekends, I would seriously consider going somewhere else. Even the lesser-visited spots in the park are going to be jammed. And it's only going to get worse. Under study is everything from reservations to mandatory shuttles.
However, if you can swing a visit on a weekday (especially Tuesday through Thursday), you'll be able to get in and out relatively easily at any time of day, and enjoy a lot more of the place to yourself.