That must be some toxic paint if the grass is unwilling to grow on it.
asphalt may crack easily over just a couple of years without maintenance. These cracks can either uncover soil beneath or capture dirt within them where seeds can eventually grow. I've seen grasses and weeds growing from small potholes left unmended and the growth will open up the cracks even further to the point the surface simply crumbles. Good road marking paint can take much longer to wear down and any dirt collected on top would be washed away by rainfall and collected by the cracks in the asphalt.
