I think recordings default to hts/video, as that's the default user/group of the tvheadend process - not sure how you have it configured, though, but that's where to start.
Try su-ing to the user under which you run tvheadend and creating a file in your target directory. A chmod of 777 on the directory should solve it, though, so something's still not right there... I presume you're creating sub-directories by channel/title/date or similar? Is it that step that's failing?
You could try changing the target directory as well, especially if your target drive is on removable media (which it will be on a Pi) as write permissions may be controlled at mount time? (guessing)
I'd start with su and go from there, as it's not tvheadend per se, it's an OS/UID/GID issue, I think.