You have a fault in the original wiring - the remote start system is "bypassing" that fault, allowing them to work when the starter is active. Touching the brake pedal deactivates it, which is why they go dead.
But the fault is probably due to improper installation of the system, so if that's what Ford finds, have them take pics & then send the bill (plus some $ for your inconvenience) to the installer.
BTW
The word is "
signal".
