but, if anything, the opposite is true. If I keep dumping a constant amount of heat onto a material it will get hot enough to cool itself.

So a piece of aluminum will be cooler than a piece of iron. The Al might transfer more heat to my fingers than the steel... or it might not (that depends on many more factors than just the conductivity).

That's not an analogy, it isn't explaining something in terms of something else.

Analogy : a comparison of one thing to another for the purpose of explanation.

https://www.lexico.com/en/definition/analogy

That's exactly what your parent said.

Parent: "That's not an analogie, it isn't explaining something in terms of something else."

You: "Analogy : a comparison of one thing to another for the purpose of explanation."

Well, the parent says this isn't an analogy, as it's not "comparing one thing to another for the purpose of explanation".