The forward completed the full game while his counterpart was withdrawn after the hour mark.
Key Moments
Antoine Semenyo's long throws created goals for David Brooks and Justin Kluivert.
Chelsea's goals came from Palmer and Fernandez during a chaotic opening 45 minutes.
Enzo Fernandez missed the best opportunity of a goalless second period.
Bournemouth forward Antoine Semenyo, reportedly close to a big-money transfer, was at the heart of the action, providing two assists and conceding a penalty in a dramatic 2-2 stalemate against Chelsea.
Despite being on the verge of a lucrative switch, the 25-year-old attacker started for his club. He utilized his long throw expertise to great effect, creating scoring opportunities that were finished by Brooks and Kluivert.
The deadlock was broken early when Semenyo's first long throw caused havoc, allowing Brooks to score from close range in the sixth minute.
The home side equalized and went ahead following a foul by Semenyo on Estevao in the box. The match official awarded a penalty after a review by the Video Assistant Referee system.
Palmer calmly dispatched the penalty, placing it perfectly into the corner even though the keeper dove the right way. Fernandez made it 2-1 with a stunning strike into the top corner midway through the first half.
Bournemouth hit back almost immediately, as another Semenyo long throw was turned in by Justin Kluivert from close range just four minutes later.
Chelsea created further opportunities, but Bournemouth ended the first half strongly, testing goalkeeper Robert Sanchez with numerous attempts.
The home side's boss reshuffled his team at the break, bringing on James and Neto for Acheampong and Garnacho.
Early in the second half, a handball appeal against Malo Gusto was dismissed by the VAR official. Petrovic, once of Chelsea, made a fantastic stop to keep out Estevao's powerful drive from the edge of the area.
Fernandez had the clearest opportunity to win it late on, but he skied his effort from a promising position inside the penalty area.
Moments earlier, Palmer was substituted for Joao Pedro, a change that was met with disapproval from some home supporters. Despite growing dominance in the latter stages, Chelsea could not find a decisive late goal.
Deep into added time, Unal connected with an Adli cross but sent his volley over the bar from a central position.
The result means Chelsea have managed just one victory in their last seven league outings. Bournemouth's winless streak now stretches to ten matches across all competitions.
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