The Rams were not only heavy favourites to beat Carolina but are one of the favourites to win Super Bowl 60 on 8 February.
Despite coming into the post-season with a losing 8-9 record, the Panthers had home advantage having stumbled to the NFC South title.
The Rams, who finished the regular season 12-5, looked on course for a straightforward passage to the Divisional Round after Puka Nacua scored the opening two touchdowns.
But the Panthers regrouped and cut the score to 17-14 at half-time, before tying the game in the third quarter.
The final quarter produced four touchdowns, with the Panthers twice going in front, only for the Rams to reply each time through Kyren Williams and Parkinson.
Both scores were from passes by Rams quarterback Matt Stafford, the favourite for this season's Most Valuable Player award, who threw for 304 yards, three touchdowns and one interception.
Panthers quarterback Bryce Young threw for 264 yards on his play-off debut, plus one interception, while throwing for one touchdown and rushing for another.