NOTE: My answer is US specific
Not only is there no need to operate from an airport, it is recommended not to, if there are aircraft operating from the airport. Paramotors operate (in the US) under FAR Part 103.
103.13 Operation near aircraft; right-of-way rules.
(a) Each person operating an ultralight vehicle shall maintain vigilance so as to see and avoid aircraft and shall yield the right-of-way to all aircraft.
(b) No person may operate an ultralight vehicle in a manner that creates a collision hazard with respect to any aircraft.
(c) Powered ultralights shall yield the right-of-way to unpowered ultralights.
In addition to the above,
103.15 Operations over congested areas.
No person may operate an ultralight vehicle over any congested area of a city, town, or settlement, or over any open air assembly of persons.
as well as
103.17 Operations in certain airspace.
No person may operate an ultralight vehicle within Class A, Class B, Class C, or Class D airspace or within the lateral boundaries of the surface area of Class E airspace designated for an airport unless that person has prior authorization from the ATC facility having jurisdiction over that airspace.
It is common to operate a paramotor from (launch from and land on) a park or other cleared field, as long as you don't launch over a large number of people. In my area of Virginia, there are some paramotor pilots who use a public park and launch themselves along an edge, so as not to fly over on-going soccer games.