One physical gesture sits under this root: you take something whole and you open it out. Done to bread, that gesture is slicing (a פְּרוּסָה is a slice). Done to a map, a flag, or an army, the same gesture is spreading / deploying. The action-noun פְּרִיסָה happily means both "slicing" and "deployment", because Hebrew sees them as one move. The famous noun פְּרָס, a prize, sits beside the verb as a likely cousin: a reward spread out before you.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
A single lexicalized verb (pa'al) anchors the whole family; the spreading/deployment senses also surface through the military reflexive noun הִתְפָּרְסוּת. Hebrew did not lexicalize the other binyanim for this root, so the nouns carry the range.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.