Abscess of gums can be because of many reasons. so it is very difficult to say who can handle it.
Basically any dentist
who is qualified can handle an dental abscess
If you need an specialist then in specific terms if it is from the periodontal origin then periodontist can handle it well.
If it is from the tooth origin then endodontics can handle it well..
My suggestion is go for diagnosis first and then treatment.