The Cherries, with four academy graduates in their starting line-up and seven more on the bench, twice took the lead before being denied victory by Callum Robinson's well-taken second half goal.
Parker said: "Welcome to the Championship! I am immensely proud of the team and the players.
"That was a football match against a very experienced side who constantly looked to put us under pressure with their physical presence and playing for corners and set plays.
"We had a young, young team out there and I am immensely proud of what they stood up to. It can be tough sometimes and it was tonight.
"We are missing a few players and hopefully those boys will be coming back in the coming weeks to strengthen us and we know we still need to add to the squad.
"I have been nothing but impressed with these young boys. Their experience of the Championship will not become more physical and more brutal than it did tonight.
"I thought we showed our qualities at times and showed how good we can be.
"I'm ecstatic with what the young boys have done but it is clear we need a bit more strength in depth.
"It's not through a lack of trying. There is dialogue, almost by the hour to try and get players in but it is a tough market."
Bournemouth took the lead in the 12th minute as summer signing from Brentford Emiliano Marcondes immediately endeared himself to the Cherries faithful.
Philip Billing released Jaidon Anthony down the left and the youngster pulled the cross back for Marcondes to fire past David Button from 10 yards.
Bournemouth had to wait until just seven minutes after the restart to restore their one-goal advantage.
Zimbabwean defender Jordan Zemura made a good break down the left before drilling in a low cross which Button could only claw into the path of Billing and the Dane side-footed home from 15 yards.
Robinson got the Baggies back on level terms for a second time in the 67th minute, expertly steering the ball into the far corner on the half-volley after sliding in to meet Townsend's cross following a flowing team move.
New manager Valerien Ismael said: "For an opening game it was entertaining. We gave a lot of answers.
"In the first half we were on top with our intensity, but Bournemouth were clinical with the goal they scored.
"Throughout the game we showed a good mentality to come back from behind twice.
"Bournemouth were strong as we expected with a lot of possession and clear build up.
"I saw a lot of positive points tonight and we just need to improve our confidence and clarity."