An odd thread.
And an odder - and somewhat disingenuous - thread title which has little to do with the topic of the thread.
Most of the replies have focussed on the fact that the OP seems to have disregarded the wishes of the parents in how he passes the time with the kid, (by showing her TV) but this, to me, is not the major issue, although it is of some concern.
Of greater concern to me is the fact that the parents appear willing (if not desperate) to offload their child on someone else several times a week. Wacky or not, I call that taking advantage of someone else while being grossly irresponsible about raising and supervising your own children.
If you don't want kids to engage with the modern world, or know about it, or watch TV, it is on you to ensure that you are around to enforce it, rather than asking someone to mind your kids three or more times a week.
In this instance, the father strikes me as someone who does not want the actual responsibility for raising the children (because, raising children is difficult, demanding, sometimes tedious - especially without TV - and occasionally tiresome) but wishes outsource the actual childminding to others - setting rules - without being prepared to take the time and trouble to do so himself.
Asking anyone else to mind your kids three times a week is an imposition; it is something to be asked as a favour occasionally, not a right you can demand - when you must attend a function somewhere else, or a hospital appointment, or a play; not a nightly occurrence.
And I will admit that I am surprised that the issue of payment has not come up before now, given that the time asked, or demanded, or requested of the OP has been quite excessive.
To my mind, it is not just the OP who may be viewed as "wacky" (which is what he said his neighbours called him - what does that say about them, that they are prepared to give their child, for days and nights on end, to someone they openly describe as "wacky"?), but also, very much the parents of this chid, especially the God-loving father.