If you want parents to be able to see the shopping list, you can give the ‘manage awards’ privilege. As long as they don’t mess with anything, they can go to Achieve > Shopping List and see exactly which badges are showing up on the list. An award will only show up on the list if there is a date on the “completed on” field and it is not marked as purchased. If you don’t have a date in that field, it won’t show up on the list. If you or the parent marks the checkbox for it being purchased, then it won’t show up on the list.
Adding an additional feature or privilege won’t change that behavior - the data still needs to be entered in correctly to display correctly. If a girl finishes an award or patch, it needs to be marked as completed. Once the badge/patch is purchased, you need to mark it as purchased. Once it has been given to the girl, you fill in the “awarded on” field. Once it has been recognized in front of the troop, you can mark it as recognized. (In many cases, this is the same thing, but not always.)
Alternatively, if you don’t want to give them the “manage awards” privilege, you can show them what to look for on the achievements page. If the data has been filled in properly, there will only be a date in the “completed on” field, nothing in the “awarded on” spot, and no extra symbols or icons next to the achievement.
If parents are entering in data and marking thing without knowing what they are doing, it is going to mess things up. If you would like parents and scouts to be able to submit awards progress for review by leaders (allowing them to enter in the dates that things were worked on without giving them access to the other stuff, like accidentally marking things as purchased), then you can turn on the “provisional progress” setting and remove the “manage achievement records” privilege from the parents.