Late September, early October is nice, although this is well enough known that the parks stay pretty busy, it is the most popular time for visitors flying in for RV circle tours, from Europe and the eastern U.S. Mid-October gets a huge influx of traffic whichever week the UEA has its lobbying visit to the state legislature, creating a week-long school holiday at a time of the year when southern Utah weather is most pleasant. This year it is this week.
After mid-October, it depends on your weather expectations. The area gets a long cold winter, but outside the mountains it is a mostly dry winter, and cold means the parks are not so crowded. I've been there in early October, great weather tending to be warm, but thunderstorms possible, and in May when it was moderately warm at lower altitudes, quite cold up in the passes.