Don't forget that the gates at the end of the Clayton Square 'cut through' are closed out of shopping hours, which adds a few minutes onto the walk between Lime Street and Central as you have to loop around either end of the precinct. (Or they were on a Sunday morning once, so I imagine that's the general situation.)
Personally I always use South Parkway for connecting, if only to avoid the scrum at Central during what is loosely described as peak time (i.e. anything after 14:00!).
You don't need to use Clayton Square at all. Use the side exit at Lime Street, adjacent to the arrival platforms usually used by LM/LNW trains.
Down the short slope, turn right, down another short slope, then cross the road at the traffic lights, continue past the bus shelters, then turn left (into Great Charlotte St.); turn right at the end of that street (opposite McDonalds) and you will see the entrance to Central on the left after about 100 - 150 yards.
When I was younger & fitter, I used to be able to make connections in about 6-7 minutes - including booking another ticket at Central.
Your main problem is punctuality of your arriving service. Even if you are on time at South Parkway, trains can sometimes get stopped outside Lime St., waiting for a platform to become vacant.