It is well known that the bronchial muscles have only parasympathetic innervation. But they still express $\beta_2$ receptors. How does this system still cause bronchial relaxation? What is the exact mechanism?
Good question, but first I'll address a note in the comments.
In bronchial smooth muscle, do β2 receptors receive sympathetic innervation?
Canning and Fischer (2001) claim that bronchial smooth muscle do receive sympathetic innervation. But it would still be correct to say that β2 expression plays little role in regulating smooth muscle tone in the human airway under normal conditions.
How do β2 receptors cause bronchial smooth muscle relaxation?
You already know that this receptor is associated with bronchial relaxation. An article by Billington and Penn (2003) claims it works through Gs coupling. This binds adenylyl cyclase causing it to form cyclic AMP which activates protein kinase A (PKA) which carries out many more functions.
WHY are β2 receptors present on bronchial smooth muscles considering [minimal sympathetic] innervation?
Circulating chatecholamines released due to emotion, exercise and the like. Billington and Penn (2003) note "prophylactic bronchoprotection" as the central role for Gs-coupled receptors in airway smooth muscle.
Why ELSE are β2 receptors present on bronchial smooth muscle?
β2 receptors are important in regulation of airway smooth muscle cytokine/chemokine synthesis, and inhibition of airway smooth muscle growth (DNA synthesis/cell proliferation) i.a. (Billington and Penn 2003).