Liverpool South Parkway isn't readily accessible from Preston solely using National Rail services. The NR website suggests travelling by Warrington but that involves a 25 minute walk between Bank Quay and Central stations in Warrington.
Liverpool South Parkway is served by MerseyRail. From Liverpool Lime Street (where trains from Preston terminate), hop on a Merseyrail train, for just one stop, to Liverpool Central. Then change to the other line, for 5 stops, to Liverpool South Parkway.
It would make more sense to take the train from Preston to Liverpool Lime Street, then take the Airlink 500 bus direct from there to the airport. Buses run every 20 minutes during Monday to Friday daytimes and every 30 minutes at evenings and weekends. The fare is �2 each way.
Chris