Looking for design pattern: Method overriding without subclassing

Consider the following scenario: I have a class A composed of various parts Part 1, Part 2, etc., and I want to extend the functionality of A by subclassing.

The issue: if I tried to override the behaviour of the individual "Parts" (is there a better term?) the same way, I would have to create one subclass for each Part that changes, which gets messy and confusing quickly.

In a nutshell: I want to override class methods without resorting to subclassing.

I thought of a solution, but I'm unsure if it is a good idea:

Suppose I want to override Part::Method. Instead of using the inheritance mechanism provided by my language, I could "fake" an overriding capability by:

• passing the Part class a function MethodOverride (e.g. as an anonymous function), and
• having Part::Method call MethodOverride

I can see two downsides to that approach:

• it ignores the (perfectly capable but impractical) inheritance feature of the language, which may have unforeseen side-effects
• access non-public members of Part becomes a hassle

My question(s): Is there an established design pattern that solves the described issue in a similar (or better) way? Are there any downsides I missed?