When children want to feel important and loved confident by their parents, do they want to be accepted/ approved/ acknowledged/ recognized by their parents?
Accepted. It does not sound well that parents approve their child. While a parent can acknowledge and recognize a child without giving enough attention to his child (which is tantamount to love and importance).
I am not sure as to the placement of the word "confident" is correct, but I will assume that you are either not a native speaker or that that was a typo, and that you mean "confidently".
With that in mind, I understand what you are describing to be that a child wants to feel both loved by their parents and important in their eyes. The only emotion I can find that fulfills both of these factors is pride. So I would say the best way of saying this is to say one of the following:
[they] want the parents to be proud of [them] [they] want to make the parents proud
However, if the original sentence structure is important, then "recognise" is probably the best option of the ones you gave. Although, I would say that "proud" and its noun "pride" are more emotively appropriate.