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I don't think you can control the interpolation used by Texture. One option might be to embed the image as a Raster primitive instead. Show[ParametricPlot[{20 + 1.4 x - 40 y, x}, {x, 0, 200}, {y, 0, 1}, BoundaryStyle -> Directive[Purple, Thick], PlotRange -> {{0, 201}, {0, 144}}, Prolog -> {Raster @ Reverse @ ImageData @ a}]] Zoomed ...

Using a combination of nearest resampling and a large size (e.g. 2000 pixels) should do the trick. a = ImageResize[Import["http://i.imgur.com/PiLKV6S.png"], {2000}, Resampling -> "Nearest"]; Show[ParametricPlot[{20 + 1.4 x - 40 y, x}, {x, 0, 200}, {y, 0, 1}, BoundaryStyle -> Directive[Purple, Thick], PlotRange -> {{0, 201}, {0, 144}}, Prolog ...

Not too hard. a = Image[Table[If[EvenQ[x + y], 1, 0], {x, 50}, {y, 50}], ImageSize -> Large]; Graphics[{Texture[a], Polygon[{{1, 0}, {0, Sqrt[3]}, {-1, 0}}, VertexTextureCoordinates -> {{1, 0}, {0.5, 1}, {0, 0}}]}]