The distributive property would require an addition inside parentheses.

What is being applied here is a property that says that -(-a) = a. I don't know that this has a universal name; it might be called the "negative of a negative property", or something like that. Or, you might just have been given a property that says subtraction is addition of the negative (perhaps presented as a definition of subtraction: a - b = a + (-b). Possibly you are expected to mention both of these together.

Can you list the properties you have been taught?

I am not sure that it is technically a fundamental property. It is certainly a theorem derivable from the definition of additive inverse, a convention of notation, and the following two axioms (or theorems: it has been almost 50 years since I studied this).

\(\displaystyle \text {AXIOM 1: } p = q \text { and } q = r \implies p = r.\)

\(\displaystyle \text {AXIOM 2: } u + v = w + v \iff u = w.\)

Now for the definition of additive inverse.

\(\displaystyle a + b = 0 \iff a \text { and } b \text { are additive inverses of each other.}\)

Now for a convention of notation.

\(\displaystyle b = -\ a \iff b \text { is the additive inverse of } a.\)

Now we can deduce the needed conclusion.

\(\displaystyle \text {By notational convention, the additive inverse of } -\ a \text { is } -\ (-\ a).\)

\(\displaystyle \therefore \text {, by the definition of additive inverse, } -\ (-\ a) + (-\ a) = 0.\)

\(\displaystyle \text {Again by the definition of additive inverse, } a + (-\ a) = 0.\)

\(\displaystyle \therefore \text {, by AXIOM 1, } -\ (-\ a) + (-\ a) = a + (-\ a).\)

\(\displaystyle \therefore \text {, by AXIOM 2, } -\ (-\ a) = a.\)

I must admit that I see no reason whatsoever for this kind of formalism during first year algebra. Bourbaki for Adolescents.

EDIT: What makes it worse is that we already have enough confusion by using the minus sign for three distinct purposes: as an operator sign for subtraction, as a negative value indicator for numerals, and as an additive inverse indicator for pronumerals. We would be much better off making these distinctions clear rather than mucking around with foundations of mathematics as an introductory topic. It is logically prior, but people who are just past arithmetic cannot possibly see why it is germane to anything. Notice, by the way, that the deduction shown above has nothing formally to do with negative numbers although it can easily be extended to them.