Caveat
Your code does not work, since EF does not serialize collections of primitive types. EF Core does have value conversions but it is unclear what you're exactly looking for. I'm going to assume you meant to store these as actual School
entities, since your question asks how to "extract one entry from a table".
For the sake of answering your question, I assume that your child should have a reference to the school entity, not a string property that's technically unrelated to the school entity itself, which would make it a question not related to Entity Framework and thus the question tags would be wrong.
I'll address both my assumption and your literal question, just to be sure.
If you need a relationship between a child and a school
From a purely database standpoint, there is no way to specify that an entity's (Child) foreign key should refer to an entity (School) which in and of itself has a foreign key to another entity (Parent). It simply doesn't exist in SQL and therefore EF cannot generate this behavior for you.
What you can do, is implement business validation on your code and refuse to store any child with a school that doesn't belong to its parent. Keep in mind, this requires you to load the parent and their schools every time you want to save a child to the database (because otherwise you can't check if the selected school is allowed for this child), so it will become a somewhat expensive operation.
However, that doesn't prevent the possibility for someone to introduce data into the database (circumventing your business logic, e.g. by a DBA) where this rule is violated but the FK constraint itself is upheld.
How you handle these bad data states is up to you. Do you remove those entries when you stumble upon them? Do you proactively scan the database once in a while? Do you allow it to exist but restrict your application's users to only choosing schools from the parent's scope? These are all business decisions that we cannot make for you.
If a child needs a school name without a relation to the school itself
At first sight, this seems to me to be a bad solution. What happens when the school's name changes? Wouldn't you expect the child's schoolname to also change? Because that's not going to happen in your current setup.
In either case, if you are looking to set a string property, that's trivial, you simply set the property. Presumably, your question is how to restrict the user's options to the child's parent's schools.
This restrictive list can be fetched from the database using the child's identifier:
var childID = 123;
var schoolsFromParent = db
.Children
.Where(c => c.Id == childId)
.Select(c => c.Parent.Schools)
.FirstOrDefault();
Note that this code works regardless of whether you have a School
entity or a list of strings - though the type of schoolsFromParent
will be different.
And then restrict your end user to only being able to pick from the presented options. Note that to prevent bad data, you should doublecheck the chosen name after the user has selected it.