PGY-1 here so what I’m about to say is mostly just from advice from my mentors.
Unlike other fields, neurology is an exception in terms of the fact that our length of training is NOT enough to comfortably practice as an attending. Which is the reason why 95% of neurology residents go on to do a fellowship, and even then they struggle and learn for several years. For this reason, many programs tend to discourage or downright ban you from moonlighting. Some programs will technically allow you to moonlight but won’t give you opportunities to moonlight internally. Setting up external moonlighting as a neurology resident is generally heavily advised against because the added administrative burden of setting up those opportunities will conflict too much with your training. I find it hard to imagine that any program will be willing to accommodate your work schedule for you to do external moonlighting, even if they technically allow it.
Getting off my high horse and answering your question directly - yes, some programs DO allow you to moonlight and DO give you internal moonlighting opportunities. But these are relatively uncommon. You can find them on residency explorer but I strongly discourage you from asking an interviewer as it’s sometimes received poorly.
With that being said, it’s much more common to moonlight during fellowship, granted that your fellowship is in an outpatient dominant specialty like Neurophys, movement, etc.
