By using the systematic approach of parent action method, we derive one Weyl-noninvariant and two Weyl-invariant actions of bosonic p-branes (p, 2) starting from the Nambu-Goto action, and establish the duality symmetries in this set of four actions. Moreover, we discover a new bosonic p-brane action (including the string theory) and deduce two corresponding Weyl-invariant formulations by proposing a new special parent action. We find that the same duality symmetries as those mentioned above exist in this new set of actions. The new p-brane actions are also briefly analyzed.