A burner head actuator for lubricating glassware molds of a glassware forming machine includes a base mount and a guide post carried by the mount. A gearbox housing carried on the guide post carries a gear train including drive and driven gears. A servomotor is coupled to the gearbox. A burner head arm is carried by the gearbox housing and coupled to the driven gear for rotation relative to the gearbox housing about an arm axis. The arm includes a burner head leveling gear train including a drive sprocket direct-driven by the driven gear about the arm axis and a driven sprocket driven by the drive sprocket via a chain. The servomotor rotates the gearbox drive gear, thereby rotating the gearbox driven gear and burner head arm, and thereby rotating the leveling gear train so that the burner head remains level as the arm is rotated about the arm axis.