A Muslim has purchased a very old disused cemetery.

Answer

If the ground is a Muslim cemetery, then it is not permissible to convert it into a playground. There could also perhaps be some Shuhada and Auliya buried there. Since it is known that the ground is a Qabrustaan, it should be cordoned off. People could still make ziyaarat, make dua and remind themselves of their own future abodes underground. A public playground is haraam even on a ground which is not a cemetery. Allah Ta'ala did not create us for idle sport and amusement which are condemned in the Qur'aan and Hadith.

If it is a kuffaar cemetery, a Muslim should not purchase such a ground despite it being permissible. Allah's ghadhab is on the kuffaar. The athr (effect) of Divine Wrath which is on such a place could gravely affect those who utilize the land for whatever purpose they wish to use it.