At Language Log, Arnold Zwicky has a post on illeism, which I found personally relevant, as my son Aran is 3, and I find myself frequently using appa (the Tamil word for father) to refer to myself ("Appa will get you juice," etc.). He rightly dismisses the complaints of parents who blame Elmo from Sesame Street.

I think many parents wind up using illeism. I know I adopted the habit self-consciously in response to my son's early difficulties with pronoun deixis, and now I find it to be a very hard habit to break, even though he seems to have largely worked out the I/you distinction.

However, I can't say that my usage, or really our usage, since my wife often refers to herself as amma (mother) led to any marked use of illeism on Aran's part. I can recall a handful of instances, but they were very sporadic, not at all like Elmo's usage. It was far more common for Aran to refer to himself with you and to one of us with I/me, much to the confusion of outsiders.