Early days we like to tick off 'walking', 'talking' etc. because it means they're growing normally. After the basics, what most parents want their children to achieve is happiness - and that's highly individual. In the case of my parents who came from poor backgrounds and had a low standard of education, for example, it was very important to them that their children did well at school/uni, got good jobs and didn't have to struggle to make ends meet the way they'd had to. Someone from a more wealthy background might be less concerned that their children get good jobs and more concerned that they follow their dreams. The parent that is passionate about sport might want their child to achieve in that direction so that they have something in common. A very sociable parent is going to worry if their child doesn't have a lot of friends. A shy one will get great satisfaction if their child achieves things that require confidence.
In all cases, it's important to parents because we're responsible for overseeing their development from 'tiny and helpless' to 'functioning adult'. If they're developing they're acheiving, and if they're achieving we're succeeding.