I don't know what is going on with your playground, but I do know what caused a similar problem elsewhere.
Over the last decade or so, the pavement of Interstate 495 northwest of Boston was wearing out quickly under the dashed white stripes used to delineate lanes. For a while it seemed like it could be coincidence, but then it became obvious something was degrading the pavement just under the stripes.
It turns out that the contractor that did the last repaving used the wrong paint. Apparently there are two predominant types of roadway paint, one for concrete and one for asphalt. They used the concrete paint on asphalt, which contains a solvent that seeped a bit into the asphalt and degraded it. The solvent was long gone by the time the problems appeared, but the degradation due to the weakened asphalt under where this paint was applied continued.
Many many asphalt roadways have had white paint applied, so we know it is possible to do this without any apparent harm. If differential thermal expansion was a problem, then we should see it in lots more places, given the very large number of test cases for this experiment.