American English generally uses "that" for restrictive clauses and "which" for relative clauses. E.g.:

> The local cafe that my dad loves has a lot of lunch options.

> The local cafe, which my dad loves, has a lot of lunch options.

But British English generally uses "which" for both:

> The local cafe which my dad loves has a lot of lunch options.

The "cannot function as a parenthetical" is essentially what a restrictive clause is, although of course the person you're replying to broke things into two sentences.