Yes, BlackVue dash cameras have a motion based parking mode feature no matter how the dash camera is powered. The dash camera will enter/exit parking mode based on the motion of the vehicle (5-minutes to go into parking mode and it exits parking mode when the vehicle starts moving/driving again). You must use the dash camera's low voltage cutoff setting to make sure you do not drain your vehicle's battery too low using when powering the dash camera in this manner (cigarette lighter adapter [CLA] or USB-C power adapter).
The USB-C power adapter that is sold for select BlackVue dash cameras requires the USB-C port be a USB-C Power Delivery 3.0 port for it to work. Some vehicles have USB-C ports in them for charging devices, but they may not be USB-C Power Delivery 3.0 ports. You would need to consult your vehicle's documentation to determine the type of USB-C ports in your vehicle.
USB-C PD 3.0 to DC Power Cable (USBC-3P).
Requires a Power Delivery 3.0 compatible port.
The BlackVue website states this power adapter only supports the following dash cameras:
DR970X Plus / DR970X LTE Plus Series, DR770X / DR770X LTE Series, DR590X Series.
Not compatible with DR970X Box Plus and DR770X Box models.
I do not see any statement on the USB-C power adapter's sales page stating it does not support parking mode. If the USB-C PD 3.0 port provides power while the vehicle is parked, it should allow the BlackVue dash camera to enter/exit parking mode based on the vehicle's motion based parking mode logic in the dash camera.
There is a BlackVue support document discussing the USB-C power adapter and their new OBD II power cable.
It does not state that parking mode is not supported. It does make a statement that it's not suggested to use this accessory with a portable battery pack since that type of battery is not well suited for high temperature environments.
In case you are tempted to use a regular portable battery to power it in your car in Parking Mode, however, please avoid doing so. The batteries made for charging phones and other electronic devices are generally not designed to work in extreme temperatures. They could represent a fire hazard in case they are left in a parked vehicle or even in a running vehicle in direct sunlight.