Maybe your parent meant that "1 + 1 = 2" is the definition of "2", in the sense that we write the symbol "2" to denote the Peano successor of "1", the Peano successor of "0", which is not a successor.
Maybe. Formally this is inaccurate as the successor function is not the addition function, and one has to prove that S(S(0))=S(0)+S(0).
Edit: it should also be noted that this equality shouldn't seem "obvious": the name "successor" here is misleading, as it automatically links to our everyday understanding of a successor as being "the next number", in which case it is indeed obvious. But formally it's just a meaningless symbol that can be changed to other meaningless symbols by certain meaningless rules. Our everyday life association of (apple) -> S(0), (apple,apple)->S(S(0)) is then a kind of a model for those meaningless things, but from within the formal system - they have no meaning.
In general I also do not like the invocation of formal arguments here, I think it misses the point. The bigger issue is that for us humans there is an obvious difference in kind between 1+1=2 and 0^0=1 - this should be obvious. Trying to hide this difference by appealing to different kinds of language trickery and formalism is an indication of doing something wrong, not of discovering anything new. The difference is still there, and we should understand why it's there, not force it to disappear (compare to the claim that every human act is egoistic in nature - what information does it really give us about the human nature?).
I did not in fact mean that 2 is defined as 1+1; you (and several others) are correct in saying that 1+1=2 requires proof for all reasonable definitions of +. 0^0 = 1 follows much more directly from the definitions in which it is true. My overall point has been articulated by arketyp in a sibling post to tomp's; like ^, + is "just a definition" which we arranged so as to line up with our intuitions.
I disagree with your assertion that 1+1=2 and 0^0=1 are of completely diferent kinds. That depends on what intuitive angle you approach them from; my intuition for ^ is that it is the iteration of multiplication, and in this context the only sensible definition is that x^0 = 1 (the unit of multiplication), for any x. The generalization to continuous bases and exponents should preserve this property.
But this is certainly a matter of taste, and certainly you'll find more disagreement here than in the matter of 1+1=2. What this tells us about human nature, I think, is not much: humans will disagree more when things get more complicated (^ is more complicated/abstract than +) and there is more room for disagreement.
