The last week of June is the best according to isitpacked.com
We are going the week of the 18th, and I think it will be okay. We've gone beginning, middle, and end of June before. The problem with earlier in June is that there are a LOT of school groups in the parks and then on select days there are grad parties that bring huge numbers of teens to the parks. Add that with local APs that are about to be blocked, and early June is busier than you would think.