You must enter the UK within the 90 days of this visa and collect your biometric residence permit (BRP) once you have arrived.
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This means that, at least in theory, an overseas national can leave the UK following a full 6-month visit, re-apply for a visa from overseas and subsequently return to the UK, even if this is just a few short weeks later.
The UK is part of the EU but is not part of the Schengen area, a visa to the UK does not give access tot he Schengen area. That is why you had to get a UK visa not a Schengen one to enter the UK and why a UK visa has no use to you to go to Paris. As already stated above, you need a Schengen short-stay visa from France.