quick version
Tag parents let you automatically add a particular tag every time another tag is added. The relationship will also apply retroactively.
what's the problem?
Tags often fall into certain heirarchies. Certain tags always imply certain other tags, and it is annoying and time-consuming to add them all individually every time.
For example, whenever you tag a file with ak-47, you probably also want to tag it assault rifle, and maybe even firearm as well.
Another time, you might tag a file character:eddard stark, and then also have to type in house stark and then series:game of thrones. (you might also think series:game of thrones should actually be series:a song of ice and fire, but that is an issue for siblings)
Drawing more relationships would make a significantly more complicated venn diagram, so let's draw a family tree instead:
tag parents
Let's define the child-parent relationship 'C->P' as saying that tag P is the semantic superset/superclass of tag C. All files that have C should also have P, without exception. When the user tries to add tag C to a file, tag P is added automatically.
Let's expand our weapon example:
In that graph, adding ar-15 to a file would also add semi-automatic rifle, rifle, and firearm. Searching for handgun would return everything with m1911 and smith and wesson model 10.
This can obviously get as complicated and autistic as you like, but be careful of being too confident--this is just a fun example, but is an AK-47 truly always an assault rifle? Some people would say no, and beyond its own intellectual neatness, what is the purpose of attempting to create such a complicated and 'perfect' tree? Of course you can create any sort of parent tags on your local tags or your own tag repositories, but this sort of thing can easily lead to arguments between reasonable people. I only mean to say, as someone who does a lot of tag work, to try not to create anything 'perfect', as it usually ends up wasting time. Act from need, not toward purpose.
how you do it
Go to services->manage tag parents:
Which looks and works just like the manage tag siblings dialog.
Note that when you hit ok, the client will look up all the files with all your added tag Cs and retroactively apply/pend the respective tag Ps if needed. This could mean thousands of tags!
Once you have some relationships added, the parents and grandparents will show indented anywhere you 'write' tags, such as the manage tags dialog:
Hitting enter on cersei will try to add house lannister and series:game of thrones as well.
remote parents
Whenever you add or remove a tag parent pair to a tag repository, you will have to supply a reason (like when you petition a tag). A janitor will review this petition, and will approve or deny it. If it is approved, all users who synchronise with that tag repository will gain that parent pair. If it is denied, only you will see it.