While indulgence in and of itself is not bad, the word is generally associated with over abundance in a negative way. I at times am indulgent with my children, but this is a rarity. They have boundaries and rules, so when they are indulged it is not a bad thing.
I completely agree with above posts that state that children need boundaries and limits -- they actually crave them whether they know it or not. On the other hand, indulging them once in a while makes them feel they have some say in the situation. For example, if I plan to make dinner, but my son suggests getting a sandwich from the local sub shop instead, I may indulge him. He feels special and likes he had a little power in the decision of what to have for dinner. If you never indulge a whim, then more and more things become a battle. Most children won't stop asking for things, and if they do then you as a parent know you have killed their spirit -- 'why bother having an opinion or asking for something?' -- the answer is always no. I think I actually have fewer battles with my kids because I surprise them with an indulgance of one of their whims once in awhile. The key, of course, is to always make sure they know that YOU know you are indulging them, otherwise you will have some very tyranical children!
Indulgent parenting is not a good thing for children. Children need clear boundaries and limits in life in order to be well balanced adults. If a child is used to having no rules and not having to answer to authority, this could really negatively affect the way the child interacts with teachers; and as they grow older, it could affect the way they respond to other types of authority such as laws and bosses in the work place.
The very connotation of the phrase "indulgent" parent implies that the parent does not set sufficient limits for the parent's child or children. As noted above, it is not good to be too strict; but at the same time, children who are not given sufficient boundaries are often ill adjusted and have poor social skills. Parenting is a difficult job, and one often feels that one is not demonstrating love for ones child if one does not allow the child a great deal of leeway. Quite the opposite is true. Boundaries are for protection more than anything else; the child who is given limits will know that he is loved and respected by his parents even though that recognition may not be immediate. The above post tends to split hairs. It is best to conclude that indulgent parenthood, in its plain and simple meaning, is not good for the child.