I too steer clear of Avent for the same reason's listed below.  There is a sterilising solution you can buy called Milton's.  It comes in a blue bottle.  Try that.

I used avent sippy cups with my first and quickly learned to stay away from it because of all the parts you have to take apart. You have to take all the parts apart so that the inside can get clean. And remember to quickly rinse it shortly after your child has used it because as you know, moisture gets trapped within the valves and mold can grow.   I suggest using a very diluted  bleach and soak the parts in there for 30 minutes or so and then washing it.

We have these Avent sippy cups for our kids as well...the valve in the lid will come apart and you can wash it no problem...soak them in hot water (as hot as you can get it that's not boiling)...and maybe use a bottle brush to finish the cleaning job.  Goood luck with it.

BTW...if you cannot get the black mouldy stuff out I would suggest you toss them.