Bolton finished third in the league last season behind Portsmouth and Derby County before losing out in the play-off semi-finals against eventual promotion winners Oxford United.
The Welsh side, meanwhile, were promoted from the fourth tier and opened their League One campaign with a 3-2 victory over Wycombe Wanderers last time out.
Parkinson's side and Ian Evatt's hosts could not be separated on Sunday, however, as the visiting manager suggested the result shows Wrexham are equal to the level of the third-tier competition.
"It was a good test and I was interested to see how we fared," said Parkinson.
"They started so quickly and we didn't quite react to that. But we grew into the game. We showed at times we have got quality. We had moments, so did they.
"It was important for us as well to show we can compete at this level, at the top end of the division. I felt we did that.
"At times in the second period, we gained some control with the ball and looked like we could go on and win it. But we had to rely on Arthur [Okonkwo] for one particular outstanding save."
Parkinson believes his faith in ex-Arsenal goalkeeper Okonkwo, who denied the impressive Dion Charles, has paid off after his successful attempts to sign him.
"I said to the owners in the summer Arthur was a key signing," Parkinson added. "He is someone who can grow with the club.
"He's not just a good keeper, he is a good character as well. Everyone saw his quality. He was immaculate."
Over 24 League One contests on home soil last season, Bolton had 15 wins, five draws, and four losses, and Evatt felt the home side should have continued that impressive record on Sunday.
Bolton were also denied a stoppage-time penalty for a shirt pull on skipper Ricardo Santos, much to the dismay of Evatt.
"I thought we deserved to win it," Evatt said. "We were the team probing and asking questions for large spells. We should have and probably could have gone ahead and we had enough chances to win it.
"For a team like them that ask you lots of questions and score a lot of goals, they didn't create a great deal. They really celebrated the draw so that shows where we are at."
The 19-year-old opened his senior goalscoring account to put Bolton on course for a hard-fought 3-2 victory over Bristol Rovers at the Memorial Stadium and ensured they will face Barnsley in their play-off semi-final in confident mood.
Evatt said: “I think that Shola is a hugely talented player and while I’m delighted that he’s now got off the mark with us, I’m certain that there’s lots more to come.
“He’s had a bit of a stop-start of things since he came to Bolton but scoring his first senior goal for us will be a great boost for him.
“This is his first loan away from United and his first sustained taste of senior football and as a young player you’re always going to have highs and lows.
“He started off great with us after arriving on loan in January and then he’s had a dip but the key thing is that he has kept his head down and worked hard and he can have an impact in the play-offs.”
Shoretire put Bolton in command with a tap-in midway through the first half and although Luca Hoole equalised soon after, Bolton’s superiority told after the break.
After Shoretire limped off with cramp, Southampton loanee Daniel Nlundulu restored Bolton’s advantage with a 20-yard drive before Aaron Morley struck with a delightful side-footed effort from outside the box.
Rovers substitute John Marquis slotted home from close range late on to set up a tense finale that saw Scott Sinclair’s stoppage-time strike hit the crossbar but Bolton held firm, much to Evatt’s delight.
“The squad is in a good place, the players have lots of confidence and I have a massive amount of belief in them,” Evatt added.
“We have to prove what we can do in the play-offs but now I feel we’re coming to the boil at just the right time.
“We have the hunger and determination to attack the play-offs and we’ve been working tirelessly behind the scenes, so we’ll be ready.
“But now isn’t the time for pats on the back, it’s about getting our heads down and working hard because we can do it in the play-offs if we give it our best shot.”
After taking part in a post-match lap of honour with his players and staff, Rovers boss Joey Barton was quick to look forward to next season.
“I’ll be aiming to build a squad that is capable of challenging for a place in the Championship because that’s what the fans deserve,” Barton said.
“The support we’ve received from the Gasheads has been sensational but special mention must go to my players who have worked ever so hard all season.
“I know the enormous sacrifices that they have made this season to help establish ourselves in League One.”
A free agent since parting ways with Major League Soccer side DC United at the beginning of this year, Morrison is eager to return to regular football action, sparking the interest of clubs across the English Championship.
Sheffield Wednesday, currently grappling with the challenges of a survival fight in the Championship, sees Morrison as a potential boost to their attacking options. The free agent status of the Jamaican international has also piqued the interest of League One promotion hopefuls Bolton and Peterborough, both aiming to strengthen their squads for the crucial final weeks of the season.
Having made his debut for Jamaica in November 2020, Morrison showcased his prowess by scoring his first international goal in a crucial World Cup qualification match against Honduras in March 2022. However, despite his impressive international outings, the 31-year-old has been sidelined from the Reggae Boyz squad since March 2023 due to a lack of club football activity.
Should Morrison secure a contract with any of the three English Championship clubs currently courting him, it could potentially reignite his chances of returning to the national team. The window of opportunity, however, seems tight, with Jamaica's Reggae Boyz gearing up to face the United States in the CONCACAF Nations Cup semi-finals in late March.
The possibility of a move to Sheffield Wednesday adds an intriguing layer to the narrative, considering Morrison's past stint with their cross-city rivals Sheffield United during the 2019-20 season.
With a football journey that has taken him through clubs in six different countries, Morrison stands at a crossroads, contemplating a return to the league where it all began for the talented midfielder, or the prospect of venturing into League One for the first time in his career.
