Yes they can marry you in a church outside of their parish!! The vicar at my church in travelled to a different county to marry a couple that regularly attended his usual church but ended up changing wedding location to where the bride's parents lived. They really wanted him to marry them. But you'd have to talk to him/her and ask them to liaise with the other church.
Not sure about the registry office blessing though...

My grandfather is a Vicar and he is marrying us at our local church. You just need the permission of the Vicar at the church. My grandad has travelled hundreds of miles in the past to do weddings funerals and baptisms lol. And unfortuantely you can't have any religious content at a civil ceremony, not even a hymn or prayer or anything.

I found out for a civil wedding, be it register officer or other venue, then even the entrance music etc can not have any religeous related words like god, heaven etc. x