Dilshan (117) and Paranavitana (125), following on from where they left-off in the win over county side Middlesex, responded to the challenge of following-on by taking Sri Lanka to 392 for six at stumps on the third day of four at Derby on Saturday.
That left the tourists, who started the day 227 behind, with a lead of 165 at the close and so leaving the Lions with plenty to do if they were to force victory on Sunday's final day.
Sri Lanka captain Dilshan scored in typically brisk fashion, striking two sixes and 17 fours in a mere 110 balls.
Left-hander Paranavitana proved an ideal foil, facing 216 balls in an innings featuring 18 boundaries.
Meanwhile, a scan on Sri Lanka spinner Ajantha Mendis's Achilles problem has revealed no obvious injury.
England are due to name their Test squad on Sunday and Steven Finn did his prospects no harm in a lively afternoon spell that yielded two wickets for the Middlesex fast bowler.
Both Eoin Morgan and Ravi Bopara, vying for the Eng