The correct sentence is: <em>Their </em><em>mother drives them to the park every day during the summer. </em>
Explanation:
The word <em>their </em>is a possessive pronoun, used to refer to possession when there is a third person plural "owner". In this case, we can infer that the original subject would be "the children", "the kids", or something of a sort. The pronoun <em>their </em>is modifying the noun <em>mother</em>.