Because there's no written agreement things are a bit more fuzzy.
Basically I think it boils down to this: how would the courts view the situation?
Typically when we rent to multiple adults, each adult is on the rental agreement, which means each adult is 100% responsible for the lease. We don't allow sub-leases to friends, if a friend moves in, they go through the application process, a new lease is signed with both names on the lease, etc.
So we have a situation like what I said where both people are on the lease and 100% responsible, they are both treated as tenants. Or we can have a situation like you have were lets say there was a signed agreement with the dude and then the law you posted would clearly apply; she's not a tenant and has to evict.
Now comes the fuzzy part.
There's no written agreement, there's only a verbal agreement. So the question becomes was the verbal agreement extended to include the girl as a tenant as well? Rather would a court decide that? If so then the law you posted doesn't apply.
So if I understand it correctly, they gave her the 6 week notice shortly after he left? How soon after? If it was pretty quickly (a few days) then that supports the side of treating her like a non tenant, so that supports there being no verbal contract with her as a tenant. If it was too long then a court might decide that she was treated like a tenant and in the absence of a clear agreement actions speak, and if they ever spoke and agreed that she was the new tenant, then she's a tenant.
I'm just trying to separate the issues out. Because what happens if she doesn't voluntarily leave depends on if she's a tenant or not, and since it's a verbal agreement you're kind of trying to guess which way a court will decide. I personally assume a court will side with the tenant (or girl even though she's not a tenant), because they are heavily slanted that way
So the rent for July has been paid by him correct? She's never paid any money for rent directly?
If so and the notice was given to the girl right away, I think you can treat her as a non-tenant. In that case you can't get a bailiff into her stuff because she isn't a tenant; I wouldn't feel comfortable with sending a bailiff on her without a written agreement.
Depending on what was said to the guy you might not even be able to. When he left what was the discussion like? Or did he just leave without talking to your parents? If there was any verbal discussion along the lines of implying to agreeing to letting him move out then that may be enough to be verbal agreement to end the tenancy.
If there was no discussion along those lines, he's still the tenant, and as of today he's late on his rent.
So you can do a few different things. If the goal is to get her out, I'd probably post a 14 day eviction notice and make sure he knows about it, the security deposit is the incentive for him to do what he can to make sure she gets out in that time. Of course the portion of the security deposit can go towards unpaid rent for August until it's re-rented.
The worst case is if he doesn't respond to anything, and she stays. At that point you'd have to go to court to be allowed to remove her; changing the locks and putting her stuff outside isn't an option, not without a court order anyway. Regardless of how other people act I always act within the law.
That's my take with the way I understand it anyway, its a tough situation because there's no written agreements anyway. Maybe your parents will learn a lesson...
EDIT: The other option would be to send a bailiff after him for the missing rent, though doing so for one month's rent being one day late (assuming July was paid) probably wouldn't fly. So the eviction notice thing is still probably the way I'd go.