When you do something in an overly generous or permissive way, you do it indulgently. If your grandparents spoil you, they treat you indulgently.
You can say you eat indulgently when you dine on luxurious food and take a long time to finish your meal — you're giving yourself permission to feast. Likewise, a babysitter treats a child indulgently when he lets her stay up way past her bedtime. This adverb comes from the noun indulgence. with its variations in meaning: "kindness," "gratification of another's desire," and "yielding to one's own inclinations."