My 2003 Honda Odyssey has the Brake Lamp light on the console turned ON. This is NOT the same light which comes up when you set your Parking Brake. The Odyssey manual says one of the Brake lights should be out, but all the Brake lights are working. I am worried if this is some hidden problem the system is trying to tell me.

Hello. It seems your Honda's alarm system is malfunctioning/defective. The 'parking lights' are supposed to flash on/flash off when arming/disarming the car alarm. If the parking lights stay lit; that indicates a 'short circuit' somewhere between the alarm and 'parking lamp relay'. Good luck and keep me informed; if you want to.

It would be very unusual if all of the bulbs died at the same time. Generally speaking, each circuit is on a different fuse, but they all use the same ground. So you would have a fuse for the turn signals, a fuse for the tail lamps, and so on. You should be able to test for power and ground at the sockets for each circuit. And if the front parking lights work the rear should have power too.
You may need a repair manual with wiring diagrams to be able to troubleshoot it.

This is an electrical issue. .you may need to check fuse box for loose connections or it might be grounds or short to voltage.

check the fuse box !!

This indicated that you have a brake lamp that isn't working correctly.
best way to find out which light it is, is to locate all brake lamps in the back of your car.
Have someone press on the brakes as you(or someone else) locates the brake light that doesn't turn on.
Additional information: 2001 honda odyssey brake light bulb code is 7440 can be found at autozone,NAPA,ect.

Most likely ignition wire in steering coloumn needs replaced.

Check brake light switch

open the rear hatch. Between the tail light housing and the rear hatch opening there is a trim strip with two flush fitted flap doors. Insert a small straight blade screwdriver into the small slot on the tail light side of the flap. Pry gently inboard and the small flap will open (the hinge side of the flap is on the side of the rear hatch opening.) Now you can remove the screws that hold the tail lamp assembly in place. Finally, there is still a "spear point" clip that fits into a socket near the front outboard corner of the assembly. Gently pry with a tape wrapped screwdriver (avoids scratches) and force the assembly straight back. The clip should release and free the assembly so you can change the bulb(s).

i have replaced a few caps with the floats attached.
inplug the wire to the cap and see if the light quits blinking, if so replace the cap. dont leave it unpluged it could cause problems.
make sure its full enough of brake fluid.

you can remove the cluster and buy the bulb separate as far as the brake light it is either the float is bad in the master cyl. or you have low frt brake pads