Having been used for many years,-- of the road is broken.

part of 就是个固定的搭配,指事物中的某个部分,但没有具体指哪一个.part of something is some of it.
比如,he lost part of his foot in the accident.
这个句子意思是,这条路用了很多年,有些地方都坏了.
另外,road当然是可数名词.