First things first. You can NOT tether that seat rearfacing. It is not made to be tethered rearfacing and is extremely dangerous to do so.
If the vehicle is a 2001 model, please note that there has been a recall on this vehicle. Lower anchors may not be adequately attached to the floor pan. Dealer will re-install the nuts properly. If vehicle has not been repaired, contact Ford at 800-392-3673 (Ford Recall No. 00S23, NHTSA Recall No. 00V240)... this is straight from the LATCH manual.
If your parents vehicle has not been fixed, you will need to install with the seatbelt. EDIT: Just realized I read wrong... your parents is an 02 /03... disregard the recall info.
If the seatbelt is a lapbelt. You will run the seatbelt through the rear facing belt path and simply pull on the tail end of the lapbelt while pushing down into the seat to tighten it up. You may need a rolled towel in the vehicle seat bight (where the back of the seat meets the bottom of the vehicle seat) to achieve the correct angle for rearfacing.
If the seatbelt is a lap/shoulder belt, it will most likely be a switchable retractor. Which means that if you pull the seatbelt out all the way (sloooowly) it will switch into locking mode. When it goes back in you will hear a slight ratcheting sound and you won't be able to pull the seatbelt back out without letting the entire length of seatbelt to retract back into the retractor.
In this case, you will run the lap shoulder belt through the rearfacing path and buckle. Then pull the seatbelt out all the way to switch it into locking mode and allow all the slack to feed back into the retractor while pushing down on the child restraint. Again, you may need a tightly rolled towel in the seat bight to achieve the correct angle. This method may cause the carseat to slightly tilt because of the seatbelt pulling up on the side of the carseat as it feed back into the retractor. A slight tilt is fine.
In both cases, when checking for movement, you are checking AT the belt path. Use your weaker hand and use a tugging motion... NO reefing on the seat.
You are looking for less than one inch of movement both side to side and front to back AT the belt path. The top of the carseat (where the head sits) will move pretty freely, this is normal... only test for movement AT the belt path.
Edited to add: The rear facing angle can be anywhere between 30 and 45 degrees. A more upright install is actually safer for older kids. As long as your child has good head control (typically 5 months or so), his seat can be at a more upright angle.
You should find the switchable retractor type seatbelts in the outboard positions for sure. If the middle is a lapbelt, the installation should be fairly easy.
LATCH install:
Check your parents user manual and ensure you are using the correct lower anchors for your LATCH installation.
A tightly rolled towel may be needed to help achieve the correct angle.
Ensure that you are pushing down on the center of the restraint when tightening the lower anchor belt. If you are pushing down towards one side of the seat, that may be causing your leaning issue.
A slight tilt is okay and will not hurt the installation.
Again, make sure you are only checking for movement AT the belt path. The seat will move near the top of the child restraint and if you are testing for movement up there, that could also be causing your tilt issue.