Can you see a self adhesive strip between the door &  the molding?  They
might be held on with double sided foam tape.

If not, its probably clipped on somehow. Some clip on moldings may have
a screw through the front or back of the molding. Look inside the door
jamb for any screws inline with the molding.

There are all kinds of different styles of clips and the way moldings
fasten to them.

Some typical mounting clips have two plastic spreader like wings on the
bottom of the clip.

Its basically putting a spring tension on the the lower inside of the
molding and the top of the mounting clip once the molding is in place.
The top part of the clip generally is a grooved ear that sticks out & up
and the top part of the molding sits on it.

Once the screws are removed from inside the door, close the door. Lift
up evenly on the molding at each corner. You may feel the tesioner on
the clips get tighter when you lift the molding. Lift the molding up
about 1/4 of an inch or so till you can feel its free from the top part
of the mounting clips, then  tilt the top of the molding out away from
the door.
 
Having  some assistance would help.
It is quite common for the clips to break off at the bottom. They get
brittle after 10 + years.

The dealer should be able to show you a picture on the computer of
exactly what  your clips look like - if you have clips.

You might find Old Man Rust behind the molding once its removed.

Small studs hold the clips in place. To remove, some molding clips need
to be pushed to one side to free it from the lower stud then lift up on
the molding to free it from the upper stud. Easy to figure out if you
look at it for awhile.

Don't be surprised if you can't buy one or two clips at the dealer. Last
time I bought molding clips for momma's car it was a bag of 10 for X
amonut of $$$.

Good Luck

     Harryface    
05 Park Avenue
91 Bonneville LE, 303,555  miles