Red light on dashboard on Suzuki Escudo (Vitara)

Correct. When you turn off the engine you lose the engine vacuum to the brake boost assist. No brakes. Can't accelerate with engine off.

Maybe this link will help
https://www.autozone.com/repairguides/Suzuki-Samurai-Sidekick-Tracker-1986-1998-Repair-Guide/HEATING-AND-AIR-CONDITIONING/Heater-Core/_/P-0900c152800883e2
A link talk about doing the work
https://www.suzuki-forums.com/suzuki-sidekick-escudo-vitara-geo-x/151097-heater-core-its-not-bad-you.html

clean clear , anon, double post, and still no car named,
nor real engine.

one of these?, sorry cant guess that or the engines.... ever.-
Here are some of Suzuki's most famous car models:

• Suzuki XL7
• Suzuki Kizashi
• Suzuki Escudo
• Suzuki SX4
• Suzuki Wagon R
• Suzuki Alto
• Suzuki Cervo
• Suzuki Cappuccino
• Suzuki Fronte
• Suzuki Grand Vitara
• Suzuki Ignis

Here are some of Suzuki's list of current models:

• Suzuki Ertiga
• Suzuki Every
• Suzuki Kizashi
• Suzuki SX4
• Suzuki Swift
• Suzuki Splash
• Suzuki Solio
• Suzuki Wagon R
• Suzuki Palette
• Suzuki MR Wagon
• Suzuki Lapin
• Suzuki Alto
• Suzuki Landy
• Suzuki APV
• Suzuki Jimny
• Suzuki Escudo
• Suzuki Carry
• Suzuki Equator

Here are some of Suzuki's list of former models:

• Suzuki XL7
• Suzuki X-90
• Suzuki Twin
• Suzuki Mighty Boy
• Suzuki Kei
• Suzuki Ignis
• Suzuki Fronte
• Suzuki Esteem/Cultus Crescent
• Suzuki Cultus
• Suzuki Cervo
• Suzuki Cappuccino
• Suzuki Aerio/Liana

you must test this procedure. God bless you
When you turn the ignition ON before starting the car, the "Check engine" light is supposed to come on indicating that the engine computer (also could be called ECM, PCM, or ECU) is powered on. If the "Check engine" light does not come with the ignition ON, it's possible that there is no power coming to the engine computer (e.g., due to a broken wire, faulty main relay, burnt fuse, etc.) or that there is a problem with the engine computer itself.

If the engine starts with a good RPM but does not move when engaged in gear then it is possible that the CLUTCH is faulty and needs a change as it is slipping.
If there is no higher RPM when the accelerator is pressed then the engine needs a tuning. The electrical section for ignition must be checked specially the timing and also the fuel supply and pressure.

try these guys - http://www.rebuiltautoengines.com/suzuki-vitara-used_engines.html

A typical Mazda RF Diesel Engine requires approximately 5.5litres of engine oil if the engine oil filter is replaced as well.

whenever you are ging to park your car, just make off the engine and then start it again and then reverse it, you can easily reverse ur car now.

If everything is the same - yes I would say check with Suzuki dealership. BUT in most cases the service department and the parts department will say no because they have different part #s. But if you have a good parts man he will do his homework to get the sale.

I will assume you have the same thing as in a Suzuki Grand Vitara :
N : Neutral : use this when you tow your vehicle.
4H : Use it in normal road conditions, highway, etc.
4H Lock : Use it on offroad, snow, sand, ... etc
4L Lock : Use when you are stuck on snow or sand, BE CAREFUL, this mode is only for difficult situations, do not exceed 25 MPH, and do not use it in normal conditions.
Hope this help.