A hold-up time extension circuit (HTEC) is used to charge and discharge an auxiliary capacitor. This capacitor stores the energy required to extend the operation time of critical loads experiencing short duration failures (SDF) at the DC bus to which they are connected. This paper presents complete modeling and a control-wise approach to a parallel HTEC based on the bidirectional buck-boost converter, which operates in Boundary Conduction Mode (BCM) with a variable switching frequency. The circuit permanently regulates the voltage of the auxiliary capacitor as well as the voltage of the DC bus during SDF, which is uncommon in industrial versions of HTEC. Enforcing the operation in BCM allows a reduction in the size of the inductor in the converter without requiring additional control circuitry. The entire behavior of the proposed HTEC, in all its operation modes, was analyzed theoretically and validated using simulation and experimental results, showing the potential of the circuit to be used in real applications.