To restate the question in probabilistic language, each of the $n$ chldren's parents independently and with uniform probabilities chooses a $k$-element subset of $[n]$; say $X_i$ is the choice of child $i$'s parents. You want the probability that there is a successful assignment of parents to slots, corresponding to a permutation $\pi$ of $[n]$, such that $\pi(i) \in X_i$ for all $i$.

I don't know how to answer this question, but here's a related one that gives a bound.
Consider a given permutation $\pi$.
The probability that $\pi(i) \in X_i$ for all $i$ is
$(k/n)^n$.  Since there are $n!$ permutations, the expected number of successful permutations is $n! (k/n)^n$.  As $n \to \infty$, by Stirling's approximation $n! \sim  \sqrt{2\pi} n^{n+1/2} e^{-n}$.  Of course the expected number of successful permutations is an upper bound on the probability of existence of at least one of them.  Thus as long as $k \ge 3 > e$, the expected number of successful permutations should be greater than $1$ if $n$ is large enough, but if $k \le 2$ it will be less than $1$ (and go to $0$ as $n \to \infty$).  For the example given with $k=4$ and $n=18$, $18! (4/18)^n \approx 11182$.

Of course, in real life the probabilities are not likely to be uniform: there will be some very popular slots and others that nobody wants.