Are PhD students allowed to collaborate with other faculty without their advisors being involved
The answer to both the title and the second paragraph of the question is the commonest answer to questions about PhD studies: "Discuss it with your advisor.". The advisor knows about relationships between your department and other institutions, and may know more than you do about the history of collaborations with them. There is a serious issue of whether you can make fast enough progress on your dissertation research if also pursuing other research. The advisor is supposed to give advice about how to proceed with your research career.

The answer to the first paragraph of the question body, about not even telling one's advisors, is that it is just not going to work. As soon as you co-author anything with the other institution, the collaboration is public. Even if your advisor does not come across it, someone will mention it to them. At the best it will be something like "I see your student user2987 is doing interesting work with Professor X."

Maybe the advisor would have liked the collaboration if it had been discussed in advance. If they find out about it indirectly, they will know you have gone behind their back. That is likely to damage your relationship with your advisor.