Parking mode, while it have been around for a long time now, it is still a bit hit and miss at least if you use motion detect or G - sensor.
And when you say budget, that probably also have a say to some degree.
Motion detect most often trigger on any little movement, so parked in a place with activity you will have plenty of "events" when you go home after work.
And even if there is a actual event in there, you have to sit thru many hours of footage to find it.
G-sensor will most often trigger, at least if you are getting a larger dent, it is not 100% sure it will trigger when the guy next to you open his door and put a dent in the side of your car.
I am pretty sure you could key the whole side of a car, even one with a really good G - sensor camera, without the camera detect anything as dragging a key or knife along the side of your car dont really mean there are and G force event for the sensor to detect.
Many cameras today will also support time lapse recording in parking mode, this mean they record all the time, but time lapse footage with 1 - 10 frames every second.
Alternative there is what i use, which are always recording but low bitrate, this also give you sound which the time lapse do not, the downside is as it record all the time you will have to detect the damage yourself when you get back to the car ( this might not happen if it is a small dent on the other side of your car )
The good thing is you can also have G-sensor on while recording like that, so that will trigger a event in that case,,,,, with the limitations G - sensor have.
My main camera the SG9663DR do have all the parking mode options, but most SG cameras are not budget cameras.
Parking mode are also something new i have gotten to test with the SG camera sent to me to test, otherwise i have not tried that in other cameras i have tested over the years, so i only have the basic knowledge to share and so are not really able to recommend any specific budget camera.
But i am sure other people will chime in later today.
Parking mode running off your car battery of course also necessitate you drive a bit every day to recharge what have been lost while parked, so if you only drive 10 minutes to work and 10 minutes home, and use parking guard all of the remainder of the day, then your car battery will not be able to keep up.
While a car battery initially charge fast the topping up is slow charge just like charging a phone, so to keep the charge up on the car battery and you use parking mode every day at work and at home, then you should drive a couple of hours every day i think to keep up.
Alternative you can put a charger on the car at home, but that is entirely up to how you live / park, and it is a hassle remembering to connect and disconnect every day.
Personally i just use parking mode for 1 hour ( timer in camera ) in my car, this cover all my shopping needs, i dont have to think about work as i am retired, and at home i use to have a CCTV camera on my 2 floor balcony door filming by car parked in the back year.
IF i should need more i can just enter the camera menu and select a longer timer duration, the max setting are 24 hours or until low voltage protection in the hard wire kit kick in.