How to store ParametricPlot in a variable without plotting the graph?

As @Bob Hanlon says in the comments you need to put in a ;

I think this is what you want

p1 = ParametricPlot[{2 + 2 Cos[t], 2 + 2 Sin[t]}, {t, 0, 2 π}];
p2 = ParametricPlot[{2 + Cos[t], 2 + Sin[t]}, {t, 0, 2 π}];
p3 = ParametricPlot[{1 + 0.1 Cos[t], 3 + 0.1 Sin[t]}, {t, 0, 2 π}];
p4 = ParametricPlot[{3 + 0.1 Cos[t], 3 + 0.1 Sin[t]}, {t, 0, 2 π}];
Show[p1, p2, p3, p4]

Note I had to retype your input. It is best if you use the formatting available in StackExchage so that you can paste and copy immediately into a notebook. You should be able to copy my version into your notebook. See use of {} in StackExchange toolbar. This is much easer for those of us trying to help you.