The weeds have germinated in the dirt that has settled into the cracks in the asphalt. The weeds need to be pulled and the road surfaces need to be repaired before more seeds germinate. This way you’re dealing with the cause rather than just applying a bandaid solution to the effect.
After a good soaking rain, pull the weeds up slowly so you get the root. Then spray a weedkiller, such as Roundup, along the cracks in the asphalt. This will kill any roots that are left, and deter further growth.