I am a member of a group that uses a SharePoint site (site A) and I have made a subsite (site B) that I would like to grant permission to a group of users. However, when I created a new group in the permissions page of site B, it also shows up on the permissions page of the parent site, site A. Is there any way to make it so a group, which will have automatic request access, has access to site B but not site A?
When you create a SharePoint group, the usage of this group is not limited to the site where you created it. In other words you are able to use the group on all sites, and this means that you will see the group at the Site Groups page at every site. This does however not mean that you need to give access on all sites or that the group actually has permissions to view the site.
Using the site permissions page, you are able to break inheritance with the upper site and define your own permissions. Only the groups on the Site Permissions page have access to the site.
I'm guessing you mean in the UI?
By default all SharePoint sites inherit its permissions from the rootweb(the first web in the site collection). Also when you create a group anywhere in sharepoint it will store this group in the root site, but it will have no permissions there so don't worry. This is so a SharePoint Group can be used anywhere on the site collection. This does not span over different site collection though.
You can stop this inheritance by going to site you want to stop this inheritance in and clicking "Site settings" -> "Site permissions" -> "Stop inheriting permissions" from the ribbon. Now the site has unique permissions and you can add permission to your sharepoint group here without it having access to the other site. this works the same for lists/folders/documents.
Read more here.
You can take away all (but the system
Limited Access) Permissions from the Group in the Parent site A.
That way users can do nothing in the parent site A.
Limited Access (which you can't unset) is needed to give them access to Subsite B
They will know site A is there, but have no access to its contents.
If you do not want them to even have knowlegde about Site A you have to move Site B to the toplevel.
further reading: http://en.share-gate.com/blog/what-are-sharepoint-limited-access