Yes, parallel to the carriage movement. Maybe better described as fuzzy light (as compared to the rest of the print) bars.
Ok, I am just trying to understand and think along with you as I never experienced something like that also.
It sounds like the printer is printing "ok" when the carriage goes, but doing a poor job when it comes back. I would try this (without any solid reason behind it):
1) Try to print in different quality mode, including "highest detail" and such to see if the problem happens when the printer prints in only one direction.
2) I would remove and clean the heads, clean the carriage e reset the heads. After that the printer will run the head alignment and after that I would do a paper advance calibration for the current paper.
If that does not solve it you may be facing another hardware failure.
Good luck.