Hello : )
If i understant what you want right,you dont need lookup, you just need to filter the document,
using that ids array as creteria.
This means that the intersection parents with ids should not be empty (they must have at least 1 common member)
For intersection you can use $setIntersection and then you need the result to have $size>0.
In case you really need $lookup,for example ids come from another pipelike before $lookup,
you can do the same inside the $lookup in the $match.