Area@ImplicitRegion[x + y^2 >= 0 && x + 3 y^2 - 2 <= 0, {x, y}]
(* 8/3 *)

rgn = DiscretizeRegion@ImplicitRegion[x + y^2 >= 0 && x + 3 y^2 - 2 <= 0, {x, y}];
Area[rgn]
(* 2.66664 *)

eqs = {x + 3 y^2 - 2 == 0, x + y^2 == 0};
xs = First@Solve[#, x] & /@ eqs;
pts = Solve[Reverse@eqs, {x, y}];
NIntegrate[Subtract @@ (x /. xs), Evaluate@{y, Sequence @@ (y /. pts)}]
(* 2.66667 *)

x == -y^2
x == -3 y^2 + 2

Solve[-y^2 == -3 y^2 + 2, y]
(* {{y -> -1}, {y -> 1}} *)

Plot[{-y^2, -3 y^2 + 2}, {y, -1, 1}, Frame -> True, Axes -> None,
 Filling -> {2 -> {1}}]

Integrate[-3 y^2 + 2 - (-y^2), {y, -1, 1}]
(* 8/3 *)