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From the documentation it looks like sage.plot.animate.animate has parameters xmin, xmax, ymin, ymax:

b = animate(springs, xmin = 0, xmax = 11, ymin = targetYMin, ymax = targetYMax)

Doubled checked the source code and there are no z min/max parameters.

If this doesn't work for you and you need to stabilise the z direction as well, you should use boundary geometry such that this geometry lies at the minimum and maximum bounds of the x, y, and z axes.

Here I'm adding a sage.plot.plot3d.shapes.Box with opacity 0.1 but it might work with opacity 0:

xRange = 11 - 0
yRange = targetYMax - targetYMin
zRange = targetZMax - targetZMin

springs = [
    (
        parametric_plot3d(
            [x, yy(x,t), zz(x,t)],
            (x, 0, D(t)),
            plot_points = 400,
            thickness = 2
        )
        + Box([xRange, yRange, zRange], opacity = 0.1).translate(xRange/2, yRange/2, zRange/2)
    ) for t in sxrange(0, 2, 0.2)
]