Require Component for Parent

Is it possible to use [RequireComponent(typeof(Something))], except make it require that a GameObject’s parent has that component, instead of the GameObject itself?

Not directly.

What you can do is check if the component exists, then take some action. Action could include adding the component, deleting the child, or throwing an exception.

if (!transform.parent.gameObject.GetComponent<MyClass>()){

    // Add the component

    // Destroy this object

    // Throw an error
    Debug.LogError("Parent does not contain MyClass component", transform.parent);

Edit: You could go one step further and make this happen in the editor every time you add the component, which is getting pretty close to your desired functionality.

Nothing built into Unity for that, no.

However, there are several points where you could check if the parent has a component, and log errors if it does not. For example, in your Awake() or Start() functions, or using Update() with ExecuteInEditMode, or with a custom inspector.