Dandelions have such a deep tap root it is nearly impossible to pull them, and they have hundreds of hair-like roots shooting off the tap root. If there is any part of the root system left behind, they will be back. Then there are the little parachutes that blow around forever, carrying seeds of new dandelions.
There is very little defense against the seeds, since they ride the wind. The best course is to try to get all the neighbors to kill theirs too. A broad cast weed killer is not the best cure; rather a direct spray that will be taken into the dandelion systemically, thus killing the whole plant while sparing the surrounding grass.