points = Flatten[Table[{x, z}, {x, 0, 10}, {z, 0, x^2, x^2/9}], 1];

Needs["NDSolve`FEM`"]
testMesh = ToElementMesh[points]

testMesh2 = MeshOrderAlteration[testMesh, 2]

points = Flatten[Table[{x, z}, {x, 0, 10}, {z, 0, x^2, x^2/9}], 1];
Needs["NDSolve`FEM`"]
testMesh = ToElementMesh[points]

In[]:= labels1d = First@First@testMesh[[2]];
First /@ Position[labels1d, 1];
cellstodrop = labels1d[[#]] & /@ %

Out[]= {{1, 2, 3}, {1, 3, 4}, {4, 5, 1}, {1, 5, 6}, {6, 7, 1}, {7, 8, 1}, {9, 10, 1}, {10, 11, 1}, {8, 9, 1}, {21, 1, 11}, {31, 41, 1}, {41, 51, 1}, {1, 21, 31}, {1, 71, 81}, {91, 1, 81}, {1, 91, 101}, {71, 1, 61}, {61, 1, 51}}

cellstokeep = Complement[First@First@testMesh[[2]], cellstodrop]

mesh1 = HighlightMesh[MeshRegion[testMesh[[1]], Polygon@cellstokeep, Frame -> True, AspectRatio -> 1/2], 0]

newmesh = ElementMesh@mesh1;
 testMesh2 = MeshOrderAlteration[newmesh, 2];
 HighlightMesh[MeshRegion[testMesh2, Frame -> True, AspectRatio -> 1/2], 0]