I know from travelling to most of our games at Anfield and Goodison over recent seasons that secure parking at the convent accessed from St Domingo Road is popular with Villa fans travelling from the north-west Midlands and Wales via the Wirral and through the tunnel. That parking facility has now been closed as the site is being redeveloped, even though it is still referred to on the official Liverpool F.C. website.
For fans travelling from that direction there is alternative secure parking at St. John's Catholic Primary School, L4 1SR, which is operated by the same people who ran the above site.
Directions: having exited the tunnel drive as previously along Scotland Road (A59) which then becomes Kirkdale Road. Fork left on Smith Street and along Westminster Road, then left into Fountains Road, and first right into Sessions Road where the school is located. Whilst the parking means further to walk, certainly to Anfield, then I understand it is easier to depart from after the match.