Parrott arrowed an unstoppable shot into the bottom corner with virtually the last kick of the game as the Irish backed up Saturday’s draw against Belgium with a hard-fought win.
The goal came after the Republic had four goals disallowed for offside.
The result stretches the home side’s unbeaten run to eight games.
Parrott’s piledriver was met with a mix of elation and relief as the 20-year-old forward, on loan at MK Dons from Tottenham, intervened to settle a game that had brought nothing but frustration for the Republic.
Up against a deep and stubborn Lithuania defence, Stephen Kenny’s side managed to put the ball in the back of the net on four occasions before the winner despite operating below the energetic levels that earned them a creditable 2-2 draw with Belgium.
Chiedozie Ogbene was twice foiled by the linesman’s flag while Conor Hourihane and Scott Hogan also had efforts ruled out.
Parrott’s stunning intervention will further boost the feelgood factor around this Irish team as Kenny prepares to switch focus to June’s Nations League triple-header against Ukraine, Armenia and Scotland.
Parrott responds to Kenny praise in style
Despite the fourth official’s board having indicated five minutes added time, the clock had ticked into the 97th minute before Parrott popped up to devastating effect.
After Lithuania half-cleared a Hourihane cross, the forward – who had earlier been denied by Lithuania keeper Dziugas Bartkus’ flying save – took two touches, set himself and lashed a rasping effort into the bottom-left corner.
It was Parrott’s third international goal, his first at home, and it comes less than a week after Kenny praised the young striker for “knuckling down” at MK Dons following a difficult spell in his fledgling career.
Long touted as a prospective Republic star after a string of excellent performances for the under-21s, Parrott’s inability to break into the Tottenham first team saw him take in loan periods at Millwall and Ipswich before agreeing a season-long stint with MK Dons.
With five goals for the League One side this season, his strike against Lithuania was a reminder of how clinical he can be, and will undoubtedly aid his quest to push himself into Kenny’s starting line-up before the upcoming Nations League.
Having held an understrength Belgium on Saturday, Kenny opted to rotate with six changes, including first starts for Wigan striker Keane and Burnley defender Nathan Collins.
Another player brought in was Ryan Manning and the Swansea full-back made an early impact when he sprung down the left and crossed for Ogbene to fire home, but the forward’s celebrations were cut short when the linesman raised his flag for offside.
That close call failed to spark the Republic into life during a drab first half and their frustration deepened after the restart when Hourihane’s effort was ruled out because Callum Robinson and Keane were standing in an offside position in front of Bartkus.
Ogbene then had a second effort chalked off before Hogan’s 95th-minute effort was ruled out.
Tottenham full-back Doherty was dangerous throughout but missed three chances, while John Egan had a header saved, leaving the stage clear for Parrott to provide a spectacular conclusion to an otherwise forgettable friendly.
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