First and foremost, if you're worried about re-parenting a class, then you're probably doing inheritance wrong. Inheritance should be used to establish "is-a" relationships, not simply foster code reuse. If you need code re-use alone, consider using delegation, rather than inheritance. I suppose you could introduce an intermediate type between a sub-type and its parent, but I would let that possibility drive my design.
Second, if you need to use functionality from the base class but extend it AND the use case calls for a static method, then you might want to consider using some external class to hold the functionality. The classic case for this in my mind is the Factory pattern. One way to implement the Factory pattern is through Factory Methods, a static method on a class that constructs an instance of that class. Usually the constructor is protected so that the factory method is the only way to build the class from outside.
One way to approach re-use with Factory Methods in an inheritance hierarchy would be to put the common code in a protected method and call that method from the Factory Method rather than directly call the base class Factory Method from a sub-types Factory Method. A better implementation might use the same technique but move the Factory Methods to a Factory class and use the constructor logic (internal now, not private), perhaps in conjunction with an initialization method(s), to create the object. If the behavior you are inheriting is external from the class (decryption/validation/etc), you can use shared methods (or composition) within the Factory to allow re-use between the Factory methods.
Without knowing the goal of your use of static methods it's difficult to give you an exact direction, but hopefully this will help.