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We present a new second-order directional derivative and study its properties. Using this derivative and the parabolic second-order derivative, we establish second-order necessary and sufficient optimality conditions for a general scalar optimization problem by means of the asymptotic and parabolic second-order tangent sets to the feasible set. For the sufficient conditions, the initial space must be finite dimensional. Then, these conditions are applied to a general vector optimization problem obtaining second-order optimality conditions that generalize the differentiable case. For this aim, we introduce a scalarization, and the relationships between the different types of solutions to the vector optimization problem and the scalarized problem are studied.