ClearAll["Global`*"]
Remove["Global`*"]

{ysol, xsol} = ParametricNDSolve[{x'[t] == x[t] + g*x[t]*y[t],
y'[t] == 1 - 2*x[t]^2 - g*y[t]^2, x[1] == 1, y[1] == 1}, {y, x}, {t, 0, 10}, {g}];

ParametricPlot[Evaluate@Table[{y[g][t] /. ysol, x[g][t] /. xsol}, {g, 0, 1,
1/4}], {t, 0, 2}, PlotRange -> {{-3, 3}, {-3, 3}},
PlotLegends -> {"g=1/4", "g=2/4", "g=4/4", "g=4/4"}] // Quiet

g=3.5
 y[g][t] /. ysol