The fact that we all accept that parents exist helped demonstrate a fallacy in an invalid argument.
While it would also be invalid to say that:
"I know unicorns don't exist because they don't grant every request."
We probably agree unicorns don't exist, thus the conclusion of the argument in this case is correct. However, the argument itself is still not valid. I can tell its not valid by applying it to something I know that does exist, and yet does not grant every favor that is requested of it.
Likewise, arguing that prayer is not always granted is not a valid reason to conclude that God does not exist. It simply does not speak to the matter one way or another.
Noting, that it was offered as an argument against the existence of God, reasonable people must conclude the argument is worthless, and that the web site has undermined its credibility.
You've made a good point, but the website makes the point that the New Testament has lines in it that explicitly state that prayer is always answered.
You've hit upon an argument I often make when someone tries to posit a creator God: Even if one existed, what evidence is there that this being still exists, or even knows or cares about what we want? If a deity or an alien civilization created all we observe here on Earth, then left, it would be the same as far as trying to interact with either one.