Стороны договора!

It doesn`t matter, who sings a contract with his hand - a party of this agreement or his representer. This is a party (сторона) who gets all rights and obligations coming from the agreement.

Construction company A - lawyer from A <-> lawyer from B - Concrete supplier B

Companies A and B are the "contracting parties" (стороны соглашения \ договора).

P.S. I`m Bachelor Degree in Law \ Unfinished Master in European Business Law