Semenyo played the entire match as Palmer was substituted in the second half.
At a Glance
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.
In the midst of advanced transfer talks, Bournemouth's Antoine Semenyo delivered a mixed performance, setting up two goals and giving away a spot-kick in a thrilling draw with Chelsea.
The 25-year-old, who is in deep discussions over a potential multi-million pound move, nonetheless featured for his current side. His distinctive long throw-ins proved a potent weapon, leading to goals for teammates David Brooks and Justin 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.
Chelsea responded and then took the lead after Semenyo fouled Estevao Willian inside the penalty area. 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.
At halftime, Chelsea's manager made two changes, introducing captain Reece James and Pedro Neto for Josh Acheampong and Alejandro 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.
The best chance of the second half fell to Enzo Fernandez in the 77th minute, but he blazed his shot over the bar from inside the box.
Palmer's withdrawal for Pedro was greeted by boos from a section of the home crowd as the manager's side sought control. Although they enjoyed more possession late on, the hosts were unable to craft a winning 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. For Bournemouth, the point does little to improve their own form, as they remain without a win in ten games.
