I wrote in detail about infanticide in Sparta in an older answer. The idea that the Spartans left imperfect babies to die in a chasm comes from a single passage in Plutarch, a moral philosopher writing in the Roman Imperial period. We have no earlier evidence that they did this, even though we have earlier sources describing Spartan eugenic policies in some detail. We have no accounts from historical (rather than philosophical) texts. It is most likely that the Spartans actually did not throw disabled babies off a cliff - or if they did, that they only started doing it closer to Plutarch's time, centuries after they had become politically and militarily irrelevant.

In any case, we get very little information about the process even from Plutarch. I cannot stress enough that the following lines are everything we know about the supposed Spartan practice of leaving babies to die:

Offspring was not reared at the will of the father, but was taken and carried by him to a place called Lesche, where the elders of the tribes officially examined the infant, and if it was well-built and sturdy, they ordered the father to rear it, and assigned it one of the nine thousand lots of land; but if it was ill-born and deformed, they sent it to the so-called Apothetai, a chasm-like place at the foot of Mount Taÿgetos, in the conviction that the life of that which nature had not well equipped at the very beginning for health and strength, was of no advantage either to itself or the state.

-- Life of Lykourgos 16.1-2

You'll notice that there is no list of criteria for selection or rejection. Even if we assume that Plutarch is telling the truth, we don't know on what grounds the elders determined that a child was to be raised or cast off. There is also no hint of an appeals process or of any reactions to the policy. Again, this is all we know; there are no hints of the consequences, no accounts of parents who had lost children, no stories of manipulation or subversion of the judgment.

All we can say in answer to your question is that this passage suggests that there was nothing a parent could do if the elders decided a child was unfit. This was not a decision of "the state" but of a council of elders representing the 3 tribes. The implication here is that the child is judged on the spot; if it is seen to be deformed in some way, it is immediately sent to be exposed at the Cast-off Place (Apothetai). As a Spartan citizen, you would be expected to accept this judgment, since you did not have children for your own sake, but for the sake of the community, and disabled children (in this eugenic theory) were of no use to the community. The notion of stoically enduring what the state required of you is central to the Spartan ideal of citizenship.

But, as I said at the start, all of this may be moot because the Spartans may never have practiced institutionalised baby-killing to begin with.