Q: Shaking while driving & sometimes while parked

It is very possible that you could have a motor mount issue because motor mounts are made of rubber and are meant to absorb any movements from the engine to make them unnoticeable to the driver. If the mount has failed it will no longer be able to absorb any movements from the engine and therefore any movements that are exerted by the engine will be felt by the driver as vibration. I would recommend having your engine and transmission mounts inspected by a professional to pinpoint the cause of the shaking and determine if they are in need of replacement

As for the coolant leak, I would recommend setting up an appointment with a mechanic, such as one from YourMechanic, to come perform a cooling system pressure test on your vehicle. This will pressurize the cooling system without the engine running which will enable the technician to hear or see the coolant leak without the interference of the engine noise and air coming from the cooling fan.