In your situation, I would first address the person to whom the email is most relevant. For example, if the email is about an aspect that the younger program director oversees, then address him/her first even though the clinic director is more senior.
If the subject matter is of equal relevance to both individuals, I usually address the person I work with more often first. Alternatively, it is perfectly acceptable to address the more senior person first if the subject matter is of equal relevance to both and if doing so is your preference (i.e., more compatible with your cultural sensitivities).
What is less appropriate (in my view) is to address the senior person first when the email is more relevant to the younger person.
This is just my own personal preference.
You certainly can write "Dear all," which is what I often do… :)