To distinguish homoiconic languages from others (where you may also "operate on the parse tree just the same as you can on regular program data"), a bit more qualification is needed though.
For example, in typical cases, reasoning about such data structures (lists in Lisp, terms in Prolog, bytes in assembly code etc.) is very convenient in homoiconic languages, and in fact I think one could rightfully regard this ability to conveniently reason about a program's abstract syntax tree via built-in language mechanisms as a key advantage of homoiconic languages.
For example, in typical cases, reasoning about such data structures (lists in Lisp, terms in Prolog, bytes in assembly code etc.) is very convenient in homoiconic languages, and in fact I think one could rightfully regard this ability to conveniently reason about a program's abstract syntax tree via built-in language mechanisms as a key advantage of homoiconic languages.