My information is commercial, not residential.
Commercial code no longer allows true flat roofs, they require a minimum of 1/4 per foot fall. The parapet/gutter decision may be influenced by your exterior material. Parapets require attention to detail so they are structurally sound. The exterior wall will be subjected to very different weather conditions above the roof line. The inside of parapets are typically roofed with the same material as the roof with a metal wall cap and fascia. The dimension of the fascia is determined by what needs to be covered up to overlap the finish exterior surface. Parapets would typically have roof drains with overflow scuppers, sufficient pitch to shed elsewhere, or through-the-parapet piping usually to a conductor head and downspout.
I would think your biggest concern would be compatibility with what is there now as you describe a 3 wall addition. Are you tying into an exterior face or are you picking up an existing roof line? If you are tying into an existing face, you need to pay great attention to flashing at the tie. Any waterproofing at the parapet should continue at the same height across the tie. Parapet walls will eliminate eave overhang and sun shading and will require good attention to water exclusion from the building envelope. If you have an existing pitched roof you should make every effort to continue the roofing system and pitches in my opinion. Many commercial buildings use parapet walls, are you sure that is the look, or did you mean a mansard-similar, but more residential?