can I get the contactpoints of the object that's colliding rather than the object I've collided with?

I have my player object colliding with an object, I know how to get the contact points of the object I’ve collided with but I need to know what part of my player model hit the object. How can I accomplish this?

A contact point is a point in 3D space, so you are not getting the contact point of the other collider. You are getting the contact point of both colliders. And the ContactPoint structure contains a reference to both colliders.

How to determine what part of the player model is hit will depend on your model. If the model is something regular, or something where you can decide “what part” based on a local coordinate, you can use Transform.InverseTransformPoint() to bring the point into local space and then use the nature of your player object to determine what part. But you might be better off using separate colliders for different parts of your player. If you need a mesh collider that might mean breaking your player apart in to separate meshes. Note a game object can only have one collider of any one type. So you may have to implement the multiple colliders as child empty game object, each with their own collider.