Emergency brake sensors and automatic brake system malfunction?

The front radar sensor fault could be triggering both issues. The red light isn't for parking only, it also warns of a primary braking system fault, in this case the RADAR unit that is part of the smart braking system seems to be causing it. Diagnostics on that system requires a scan tool so allow you to determine if it is a fault with the radar transceiver or with the circuitry. However, it can be as simple as the unit needs to be adjusted or something is blocking it. Check that there is nothing metallic in the center section of the nose below the Mazda emblem. The unit is mounted to the bracket just inside of the grille and it will have a square cap over it. Debris can get on it and block the signal. If there is nothing, then you will need to start with a dealer level scan tool to determine what faults are listed. Then if the unit needs replacement, you will need either a dealer or a well-equipped shop because they are not a simple bolt in part. They require a special testing system to aim the radar unit once it's installed and many places do not have them due to both price and the working room needed to do the adjustments. The attached is the primary set up for just that RADAR units testing. This is just one of the ADAS systems on the market and one specific testing unit (they use different tools for the various makes and models).