Consecutive P1s, P5s, and P8s are not okay to be used while P4s are allowed. What's the reason?
It's important to know the reasons behind the recommendation to avoid consecutive unisons, fifths, and octaves. If one wishes to have two (or more) independent voices, such repetition weakens the independence. If one is just doubling a single voice, there is no problem. Consecutive fourths generally do not weaken the feeling of independence between voices; such fourths generally appear in successions of first inversion chords (6-3 chords). Some older theorists also suggest no more than 3 or 4 (or some other number) of consecutive thirds or sixths should be avoided so as to keep the music sounding like several simultaneous melodies rather than just a succession of chords.
In most popular (country, rock, jazz, light classical, etc.) the only parts that usually need to be independent are the bass and the melody.
There are no such rules unless you are choosing to write within a particular style.
For example parallel fifths and parallel octaves are frowned upon if you are writing in an 18th century style (imitating Bach for example) but that is a very specific instance.
If you are composing something of your own you are free to do whatever you want. If it sounds how you want it to then that's ok.