The idea of a 'secret platform' - which would require an additional running line, and diverting any trains serving it off the main running line - does seem implausible. and any such platform, if regular passenger trains were occasionally diverted to serve it, would not remain secret for long.
The already mentioned use of the closed station at
Down Street is fairly well known now - I can't immediately think of any other closed stations that had a similar second life, but it's possible. There are other disused bits of line / disused stations / entrances to stations that have been closed for a variety of reasons, and are still used either by the underground or other organisations - while they are not open to the public, they are not really anything 'secret'
Other things that may have happened leading to rumours -
There are a few connecting lines between one underground line and another (I think I'm right in saying that apart from the waterloo & city, it is possible to get a train from acton works to any line on the network) that are not used for passenger trains. As well as sidings (and a loop at Kennington) that are used for turning trains round, some more regularly than others.
There are a few bits of underground that were built and never used (e.g. the 'fleet line' as it was then known nearly reaching Aldwych from the - now closed - Charing Cross station.) I think I have read somewhere that there are a couple of tunnels somewhere under the Deptford / New Cross area that were partly a test exercise, but would also have formed part of the fleet line (which in the early 70s was intended to go to Thamesmead)
It is possible that there are buildings that had provision for underground extensions built in to the foundations, and that these extensions never happened. There are also the deep level shelters built during WW2 with a possible view to them becoming stations on a post-war northern line relief / express line which never happened.
The idea of there being a private doorway at below ground level from underground station to basement level of the houses of parliament is fairly plausible. It's not impossible that a similar arrangement might have been considered at some big office buildings in central London.
I can also find an
online reference to Selfridges trying to get an exit from Bond Street underground directly in to the store, but this never happened.
http://www.abandonedstations.org.uk/ - if you've not seen it - is probably the best resource on abandoned / disused / never completed bits of the underground.