Back in 2009 Eric teased us that things would change in C# 4. So where do we stand today?
The classes used in my answer can be found at the bottom. To make this easier to follow, we will use a Mammal class as "parent", and Cat and Dog classes as "children". Cats and dogs are both mammals, but a cat is not a dog and a dog is not a cat.
This still isn't legal, and can't be:
List<Cat> cats = new List<Cat>();
List<Mammal> mammals = cats;
Why not? Cats are mammals, so why can't we assign a list of cats to a List<Mammal>?
Because, if we were allowed to store a reference to a List<Cat> in a List<Mammal> variable we would then be able to compile the following code to add a dog to a list of cats:
mammals.Add(new Dog());
We mustn't allow that! Remember, mammals is just a reference to cats. Dog does not descend from Cat and has no business being in a list of Cat objects.
Starting with .NET Framework 4, several generic interfaces have covariant type parameters declared with the out Generic Modifier keyword introduced in  C# 4. Amongst these interfaces is IEnumerable<T> which of course is implemented by List<T>.
That means we can now cast a List<Cat> to an IEnumerable<Mammal>:
IEnumerable<Mammal> mammalsEnumerable = cats;
We can't add a new Dog to mammalsEnumerable because IEnumerable<out T> is a "read-only" interface i.e. it has no Add() method, but we can now use cats wherever a IEnumerable<Mammal> can be consumed. For example, we can concatenate mammalsEnumerable with a List<Dog> to return a new sequence:
void Main()
{
    List<Cat> cats = new List<Cat> { new Cat() };
    IEnumerable<Mammal> mammalsEnumerable =
        AddDogs(cats); // AddDogs() takes an IEnumerable<Mammal>
    Console.WriteLine(mammalsEnumerable.Count()); // Output: 3. One cat, two dogs.
}
public IEnumerable<Mammal> AddDogs(IEnumerable<Mammal> parentSequence)
{
    List<Dog> dogs = new List<Dog> { new Dog(), new Dog() };
    return parentSequence.Concat(dogs);
}
Class definitions:
public abstract class Mammal { }
public class Cat: Mammal { }
public class Dog : Mammal { }