You are not understanding how this works. Maybe this will help
1) in the LrC catalog there is a hierarhcical list of all your KW's (Keywords - aka Tags). This list can be seen in the "Keyword List" pnael in the library module.
2) The "Keywording" panel also manages this list. If you are having problems (which you are), close the Keywording panel and ignore it as it does not give you a complete depiction of what was going on. Just use the Keyword List panel
3) In LrC you associate iamges with one or more keywords. This is a bi-directional association where one image can have many keywords and any keyword can be associationed with many images.
4) If you explicityly associate a KW with an image it implicitly also associates any parent keywords the assigned KW may have. So for example, if you assign keyword "Boise" to an image it will implicitly also consider "Idaho" as also being associated with that image.
5) Nonthing metnioned so far has anything to do with what you see in Windows File Manager, and none of this information is synchronized with the "tag" information in the actual image files on disk unless LrC is instructed to do so. This can be done in two ways. First is that you can set a preference to automatically write XMP data to disk files (Menu: Edit -> Catalog Settings -> Metadata tab). Second you can select one or more images and use menu "Metadata -> Save Metadata to File". In either case this will update the file on disk with what is in the LrC catalog.
Please use actual panel names in future replies so we know what you are referring to. Your screen shot shows the Kework List panel so I assume that is what you mean by "keyword tag panel". The screen shot shows that for the selected photo, it has Agnes explicitly assinged to the photo and Family is implicitly assigned.
BUT if I try to delete Family ( showing as a parent keyword) it will also delete all the child keywords
This is correct. If you delete a parent Keyword (e.g. "FamilY") in LrC it will automatically delete any children keywords as shown in the Keword List Panel. This is as designed and is the logical way for it to work.
Even stranger, I removed the Family tags from a small folder using a 3rd party App....but some are showing Family as a TAG
If you use anything outside of LrC to alter metadata in actual image files, LrC has no idea that you did so as you are doing it behind LrC's back. After you removed "Family" from the image file with the 3rd party app, I'm not clear where you are still seeing "Family" as a tag. If it is in the image file (using windows file manager) then the 3rd party app didn't work. However if you are seeing it in LrC, that is expected as LrC has no idea what you did outside of LrC and is showing you what is stored in the catalog
At this point, I have no idea what you are trying to accomplish? Do you want to delete keyword "Family" and all it's children? Do you want to delete keyword "Family" and retain all of it's children (and if so do you want them under a different parent or at the top level)? Etc. Please be explicit in explaining what the end goal is.
thanks.