Time machine/capsule read-only error
Read only is a fail safe mode for the backups. It means there was a problem detected with the backup and you can only restore from it. 

The system won't continue adding files since that would risk your losing even the ability to recover the files.

There is a procedure to re-attach a Mac to the existing backup once logic boards are replaced, but they assume that there isn't a failure of the backup at the same time.

You should have support from the people who replaced the logic board to help make sure you aren't just missing the steps to re-establish a backup, but if the backup were mine, I'd try backing up to a blank new USB drive and get a good backup now and then troubleshoot the Time Capsule to remedy the error causing the read-only status without risking your only backup. (Since the troubleshooting is to use Disk Utility to repair or erase the Time Capsule and then start over with an initial backup).