Is it safe to mount a pull-up bar on an exterior stucco wall?
The bracket bolts would need to penetrate the stucco and then embed nicely centered into the studs directly behind the brackets. There will be no safe mounting trying to use plastic anchors, lead expansion anchors or toggle bolts that just deal with the stucco alone. 

This pullup bar with its 32" extension out from the wall will place tremendous pullout force on the anchor bolts, especially the upper ones, when an person applies all their weight on the pullup bar. The forces would be even more if there are impulse loads such as occur if the exercise activity involves any motions other than slow steady load in the downward direction. Expansion anchors mounted right in the stucco will surely work loose and pull out. Toggle type anchors that would just bear against the inside of the stucco wall would likely crack the stucco around the mount points. 

If you do mount over studs using proper lag bolts with correct pilot hole into the wood can have the actual holes in the stucco sealed with good quality caulk or silicone. Just make sure to blow out all the stucco debris. I would fill the hole with the sealing agent before pushing the lag bolts through the bracket and screwing into the studs.