As I understand it, the letter pe becomes a fe if it has a dagesh in it, giving it the softer sound like an f. Throughout the Hebrew text, the word we say as Pharaoh has no dagesh in the pe. Does this mean the Hebrew pronunciation is or should be Paraoh?
Strong's shows that the "P" is in fact the Hebrew and Egyptian pronunciation
(one can click the speaker symbol at the top to hear it):
פַּרְעֹה Parʻôh, par-o'; of Egyptian derivation; Paroh, a general title of Egyptian kings: — Pharaoh.
— H6547 - parʿô - Strong's Hebrew Lexicon
It looks like the Greeks changed it to "Ph", and this was continued in Latin and then English:
title of the kings of ancient Egypt, often treated as a personal name, Old English Pharon, from Latin Pharaonem, from Greek Pharaō, from Hebrew Par'oh, from Egyptian Pero', literally "great house." Related: Pharaonic.
— pharaoh | Search Online Etymology Dictionary
It is the other way around. Hebrew letters with a dagesh assume the harder of their two sounds.
בּ = B
ב = V
פּ = P
פ = F
Dagesh has two different roles in Hebrew. Sometimes it is the "strong dagesh" or "doubling dagesh" and sometimes it is the "weak dagesh." In the case of the latter, the consonant sound changes. When it is "strong dagesh" it essentially indicates the consonant should be doubled, like the "b" in "jabber" is doubled--the first consonant ending a syllable and the next one beginning a syllable. If it is not a doubling dagesh, it is a "weak" dagesh, and does not double the consonant but makes the consonant pronunciation more forceful; e.g. V -> B and F -> P.
So without the dagesh, the "ph" in "Pharaoh" should sound like "f".
Wikipedia has a nice pronunciation chart for the dagesh if you scroll down a little on THIS PAGE.