Translation:The teaching staff can invite the parents.
Yes, "personnel, personnelle, personnels, personnelles" means "private" or "personal" which all are adjectives.
With "le" before "personnel", there is no doubt that "personnel" is a noun, if you remember that French regular adjectives are placed after the noun they modify. As a consequence, "enseignant" is an adjective: "le personnel enseignant" = the teaching staff.
A tutor would always be thought of as a single person, e.g. "I am a private maths tutor, but I am also on the teaching staff at Brisbane University." Teaching staff is a group of people who get paid to teach, and this sentence refers to the group. If you had said "tutors" then that would be closer, but tutor still has a strong implication of someone who works individually with a student to help them. This sentence really implies we're in a school setting (as in, a school for children), and in English, the teachers are very very rarely called tutors.
In London, my kids had lots of teachers and one tutor each year - at school! The tutor is someone follow the student and call parents if something is wrong (exemple: in high school my daughter had the Math’s teacher as a tutor as well, so she was tutor of the class - not only my daughter’s tutor, understand?!) But I understand now: Le peraonnel enseignant is always teaching staff (headteacher as well). Thanks a lot