hi..Thanks for the query..
The black /dark spots that you mention are known an post inflammatory hyper pigmentation ,which is a very common form of pigmentation to remain after any injury/disorder of skin. the earlier you treat them ,the better are the chances of them lightening.
You can use a topical cream having 2% hydroquinone
once daily on the spots, taking care not to spread it into the surrounding areas.Do this for about a month and see if they lighten ..Since these are old spots they may take more time than the fresh ones.
You can also do few sessions of TRICHROACETIC ACID(15-20%) peels on the spots only from a qualified dermatologist
.this help faster exfoliation and lightening of deeper layers of pigmentation.
You must always use a broad spectrum sunscreen with or without the above measures to prevent the spots from darkening by exposure to UV light daily..