What parameters cause sunburn?
Yes, all of those contribute to the total irradiance, which is the amount of sun power falling on a particular area, measured in Watts per square meter. You can imagine a 1m$^2$ "window" perpendicular to the sun's rays - excluding atmospheric and weather effects, the amount of sunlight passing through that window never changes. But when the sun is directly overhead, that window illuminates a 1m$^2$ patch of ground, while later in the day that same sun power is spread over a larger area. Only the angle of the sun affects how big that area is, and that is determined, as you point out, by time of day, season, and latitude. 

Because of this, the length of your shadow is a good tool to estimate the angle of the sun and your risk of sunburn. I tend to be a bit sun-sensitive, but if my shadow is longer than I am tall (i.e. the sun is below 45 degrees), I'm usually fine - that's most of the day in winter, but excludes the hours around noon in the summer. That angle will of course have to be adjusted based on your personal sensitivity.

As for your experience, I would imagine that people tend to expose more skin in the hottest month of the year (August), increasing their sunburn risk over a period with higher irradiance but less exposure (June). Sunburns also will also often show up in the afternoon because they're a result of accumulated damage throughout the day. If an individual takes X hours of exposure to burn, they're more likely to hit that mark later in the day, rather than during the peak of irradiance.