Chinups will hit mostly lats and biceps. Pullups back with very minor tricep but just for stabilizers.
A pushup would probably hit more tricep than anything you can do with that bar. Or you could literally put your legs up on something, butt in the air, and both hands behind you on a chair and go up and down as a type of tricep pushdown/dip
Do not do this, it is one of the most dangerous exercises for your shoulders, as it stresses across the anterior portion of the shoulder joint
The shoulder joint isn't a true ball-and-socket joint (like the hip joint); it's more like a golf ball on a tee. So, while highly mobile, it lacks stability offered by the surrounding bones and relies on muscles/ligaments for stability, thus, it's more prone to injury.
Dips also push the glenohumeral joint beyond the normal anatomical range of shoulder extension, stretching the joint capsule and reducing its ability stabilize and produce force, again rendering it more prone to injury. Additionally, as the humerus extends during the lowering phase of the exercise it internally rotates, adding to the potential for impingement, while also forcing a rounding of the shoulders that compromises stability within the entire scapulothoracic region.
Long story short, don't do 'chair' dips. Parallel bar dips are ok (because your arms, shoulders and body weight are all in the same vertical plane) but make sure your elbows never bend further than 90 degrees.
As far as safer tricep exercises, I'd recommend resistance band/cable pull downs. They're a natural motion that doesn't stress any body part/muscle group too much. Additionally, they isolate the tricep more so than a close grip push up, and are not awkward movements such as DB kickbacks or skullcrushers.