HVDC can provide frequency regulation during disturbances (e.g., faults) by controlling the power flow between two remote AC areas. While this action reduces the power deviation in the area affected by the disturbance, it causes a power imbalance in the other healthy AC area, leading to a frequency variation and endangering the system stability. In this work, a HVDC primary frequency regulation controlling voltage-dependent loads (PFR-VDL) is proposed, where the HVDC terminal in the healthy area influences the grid voltage amplitude to shape (decreasing or increasing) the load consumption in order to cope with the power variation required by the fault-affected area. The PFR-VDL extracts the needed energy for the frequency support, not from the generators (with following frequency deviation) but from the voltage-dependent loads in the healthy area. This work analyzes the PFR-VDL performance, generalizing it with two possible HVDC connection cases: Asynchronous connection with single HVDC line, and embedded HVDC forming a parallel, hybrid connection with HVAC. The PFR-VDL application benefits and limitations are evaluated analytically and verified by means of PSCAD EMTDC simulations, and finally validated with a large interconnected IEEE 39 bus system.