Parish won the toss and asked Thornes to bat first and opener Dean Killey made a brisk start, reaching 24 from just 19 balls before he was caught and bowled by Paul Brook.
James Wolfenden made 23 before being caught by Neil Parkinson off the bowling of Christian McBride.
Nipuna Seneratne and Jack Gibson then set about putting together a solid third-wicket partnership which proved the backbone of the win.
The pair added 73 runs before Seneratne fell for a patient 50 from 73 balls.
Gibson hit five fours on his way to 33 and Oliver Dennis also added 33 before being one of three run out victims as Thornes chased quick runs late in the innings.
Adrien Thompson boosted the final Thornes total to 197-8 with 20 off 16 balls.
McBride ended with 2-38 from eight overs and Parkinson 2-39, while Brook proved the most economical bowler taking 1-29 from his allotted overs.
Parish lost David Blakeley for 11 with the score on 32 before Parkinson and Jonathan Hellings added 68 runs for the second wicket.
Parkinson (38) and Hellings (39) were out within 12 runs of each other and wickets then began to fall at regular intervals.
Opening bowler Robert Stephenson returned late in the innings to turn the screw and claim the wickets of Andrew Dracup (18), McBride (1) and James Hardcastle (13) to finish with 4-29 from eight overs.
Eddie Hoyle was the last man to fall, run out for five in the final over as the Parish challenge ended on 184 all out.