From the Newham website:
A Blue Badge is not valid within a disabled resident parking bay. Please note the bay is not exclusively for a specific resident, but eligible for use by any vehicle with a valid disabled resident parking permit for that zone.OP, what made you think that '[your] parents have a designated disabled parking bay outside their flat on the street. The bay is registered to their car'?
Their and your misunderstanding, or just yours?
...assisted by the council IMO:
Dedicate:
devote (something) to a particular subject.
"volume four is dedicated to wasps"
cite or nominate (a book or other artistic work) as being issued or performed in someone's honour.
Whereas...
Designate is what the council have to do to to a length of highway to create a parking place in the first instance. (See s45
https://www.legislation.gov.uk/ukpga/1984/27/section/45 )
Certainly room for confusion, unless the application process and confirmation paperwork make it clear that 'dedicate' is for a purpose as opposed to a person.