the time machine, wow that is a jump, i bet even scares Luke Skyjacker.
this is a feature, called PARK LOCK out.
on most Suz, there a bypass switch, the operators guide (lost that)
sure hard to get asking for 2088 op guide at suzuki < LOL!
no thoughts, just facts, its busted. the FSM covers how to fix it.
the system works like this,
key lock. and then unlocked, it passes through the brake switch'
then to the park lockout solenoid , under shifter. and unlocks the shifter
that is so soccer mom dont drive over kids,
you know, she steps on gas, instead of brake?
the book is only 4 bucks. on line.
here it is.
its all here, all the time,
https://suzukipitstopplus.com/Products/1131-2008-sx4-service-manual.aspx
or at alldata.com all the time. 15 bucks.
see how yours works, some use cables, and some have no bypass button. newer cars , it seems this button is long, more laws, bla bla...