As regards (how I would imagine) the Abrahamic God I don't think the parent-child analogy makes any sense (at least to me anyway). [Artists impression only. Real God may differ from presented image. Other god concepts also apply]
This God defies categorisation and conceptualisation, and any attempts to do so ultimately distort His nature. Such a God is omnipotent, omniscient, omnipresent and has no beginning nor end. Attempts at categorising such a God are impossible, and it is better to express ignorance as to God rather than knowledge as it is unreasonable to think a human can imagine or comprehend anything about such an 'entity'.
Analogy is useful when it explains something in a way that people can better relate to. Analogy is very problematic though when it highlights superficial similarities, but ignores the fundamental differences that are the essence of the issue and thus obscures rather than enlightens.
The omni-max God escapes verbal description, no word that could be applied to humans could be applied to this God. If God is loving, just or merciful, these are not the love, justice and mercy of humans, as these are based on imperfect knowledge and subjective emotions.
Such a God couldn't be said to 'exist' in the sense that other things exist, even if He was 'real'. You couldn't say there was 1 God as this implies something that is finite, you could however say 'there is no other like He'. Parents exist though, and there are many others like them.
The only thing you can say about God is that He is beyond human comprehension, but everybody can comprehend a parent-child relationship. This is also contains love, hope, want, trial, error, regret, pride happiness and sorrow, none of which can be ascribed to God.
The way parents create a child, is also nothing like the way God created humans.
I'd say that there are actually no similarities whatsoever between the parent-child and God-human relationship.