This 3 year old has had huge changes in her little life recently :
Moving house
Mum back at work
New nursery
All life changing events for a three year old. It will have impacted on her sense of security. You have looked after her since she was 1 year old and she may associate you presence with mum and dad leaving her (which up until now, with everything else in her little world as she thinks it should be, will not have bothered her) ) and certainly she will equate the nursery with being left!
Crying but then being ok once in the nursery is not manipulative in a 3 year old, it is the solidity of the situation they are in at any given moment! So ...on the journey, you being there as well (not sure why you are?) uppermost in her mind is being left. In the nursery, she sees things she wants to play with etc, brings to mind that it was ok previously ...being left is no longer at the forefront of the mind, the fun memories are!
I would suggest:
*Don't go with her and dad to drop her at nursery
*See her at weekends a few times with her parents and have some fun new little things to do. Get them out and start playing, let her come to you, create new associations for her. *Her parents can be in and out of room for gradually longer periods over a few weeks.
*Look after at her house a few times, with Mum and Dad maybe out for an hour or whatever.
Don't try too hard, just be around and do things that will attract her. Avoid asking her lots of questions about her behaviour. Maybe find out some songs/books/toys they have at nursery, buy some, get them out of your bag/start singing or start playing with them. Wait for her to tell you she has those at nursery,
All of this will build her confidence and security up again.
Good luck 