Franklin's parents either inherited, or make their living as gardeners.Franklin's world is not a cashless society, so unless they inherited, they have to be doing something for work. Yet, no mention of an occupation is ever made and they are actually shown to be at home during the day when Franklin is at school. Since a large amount of that time seems to be spent working in their garden, it makes sense that it is their occupation and those fruits and veggies become food for the stores in Woodland. In the CGI Franklin and Friends series, it is revealed that Mrs. Turtle sometimes takes substitute-teaching work at Franklin's school. That's income, but it could use supplementing.
Franklin and Friends is, if not a Reboot, then a continuation of the Original Animation except where noted.Word Of God has it that Franklin and his friends have grown up a bit in the new show. That doesn't explain, though, the curious absence of certain characters like Granny, Badger, Raccoon, Aunt Lucy, Sam and Kit. (And if it weren't for a couple token appearances, one might as well include Skunk in that list as well.) Yet, aside from this and other minor issues, there is really so much continuity that it feels like the show is taking place in the same basic continuity. So it seems more fair to say that the show is a continuation of the original except where noted, rather than calling it a flat-out Reboot.
Raccoon is partially home-schooled.How else can he get away with not attending Franklin's class in about half the episodes of the original television series? (He doesn't appear at all in Franklin and Friends.)
By the time of Franklin and Friends, Granny Turtle is already dead.Granny lives in Woodland at a relatively close distance from her son's own home, yet her character never gets a passing mention in the series. Death by Adaptation is a possible explanation from this, as she suffered a serious illness before her seventieth birthday in Franklin and the Turtle Lake Treasure.
By the time of Franklin and Friends, Badger is being homeschooled, has moved out of Woodland, or has has passed away as well.Badger was part of Franklin's class in the entirety of the Original Animation, but starting from Franklin and Friends she is suspiciously absent. Her cerebral palsy may have had an influence in this as she could simply be homeschooled now like Raccoon, moved out of Woodland in order to cope with her disability, or simply perished due to the debilitating nature of her condition.