No, thanks
There’s a growing number of social networks that would ask you to give out your email password to them. FaceBook, LinkedIn are among many others.
I read some random articles on this topic and the point is the same:
NEVER. GIVE. YOUR. PASSWORD. TO. ANYONE.
This is a basic rule of privacy and you’d be screwed if you don’t follow it. So the particluar site might be perfectly trustworthy and 100% secure but if they were nice guys they would not ask this in the first place. Give them your email password and someone else will ask you the password for your bank account.
Now, next time they ask you for your password you know what you’ll say, right?