I did think about suggesting this to you but in the case of the door at your parents house it is more complicated than the one that RJ has in his photo. The damage you're dealing with is to both the rail and the stile. It is possible to patch this but you need to consider that the joint between these two parts of the door is prone to movement so you will need to do 2 patches that go with the grain of the part of the door that they are in. These patches will need to butt up to each other but not be glued to each other so that movement can occur.
I did think about suggesting this to you but in the case of the door at your parents house it is more complicated than the one that RJ has in his photo. The damage you're dealing with is to both the rail and the stile. It is possible to patch this but you need to consider that the joint between these two parts of the door is prone to movement so you will need to do 2 patches that go with the grain of the part of the door that they are in. These patches will need to butt up to each other but not be glued to each other so that movement can occur.