Home goalkeeper Mark Travers also pulled off a string of good saves late on and Parker said: "We were far from at our best.
"In the first half we were clinical but they put us under immense pressure in the second half and we had to show another side to us.
"Mark is growing, is listening and working tirelessly on the bits he needs to improve. I thought he was very good at dealing with high balls and he pulled off some good saves when he had to. It was a big performance from him.
"We have seen tonight that every game in this league is different and that sometimes you have to grind it out."
It was the visitors who made the brighter start and the Cherries took the lead against the run of play in the 12th minute from their first meaningful attack.
It was 2-0 in the 37th minute when Solanke side-footed past Dieng from seven yards after playing a neat one-two with Anthony to claim his sixth goal of the season.
Bournemouth almost added a third in the 53rd minute when Philip Billing struck the post with a right-footed shot when he should have scored.
Billing was made to pay for that miss four minutes later as McCallum's close-range shot was deflected beyond Travers by Gary Cahill.
Rangers could have levelled when defender Jordy De Wijs' 81st-minute header was brilliantly tipped over the crossbar by Travers.
Irishman Travers also had to react smartly to turn Lloyd Kelly's attempted clearance behind for a corner and prevent an own goal.
Travers then did superbly to keep out substitute Andre Gray's header from point-blank range in the sixth minute of stoppage time.
Visiting boss Mark Warburton said: "It is fair to say I'm frustrated. We started the game very well and then a player makes a mistake.
"I will never criticise a player for that but it is about how you respond and the second goal was soft. We have to learn and we have to be better.
"Once we got the goal we ended up looking very threatening and got bodies forward. I thought we had got the equaliser right at the end with the header from Andre."