Is there any kind of parser software analysis tool that will analyze C code and outputs possible vulnerabilities like buffer overflows?
"Static Code Analyzers" is the search term you are going for. I do not use them myself, so I will only list some resources that I hope might help.
You might find it helpful to take a look at this list of application security testing tools over at IT Central Station and to read through the user reviews.
All of the reviews have been written by people with personal experience using the software, and include the pros and cons and other details that the reviewers thought was important to share. You can also message the reviewers if you want to ask them for more information about the software. For example, one user commented on HPE Fortify on Demand "The static code analyzer provides views from a security perspective and it is easy to use compared to others." (You can read the rest of this review here).
Hope this helps.