The measure of possible data sets where, by standard statistical analysis, the coefficient for the squared component is zero, is tiny. There's no guarantee for a parabola, just a very high degree of certainty.
Your other point, that the parabola could be "not statistically significant," is true.
But given a strong degree of significant correlation between unemployment and the cognitive indicators, even if the dependence is totally flat for the cognitive indicators between 5 hours worked and 100 hours worked, you will still get a parabola by this method of statistical analysis.
Sigh. No coefficient is ever exactly zero, just very close [0]. I didn't think I needed to explain that when writing, "could be zero."
> you will still get a parabola
If the squared parameter is not statistically significant, the author will likely drop it from the model. In that case, we would not see a parabolic model and the paper wouldn't exist. The authors would have moved on to a different topic, or found a different dataset.
If the coefficient is so small that it is indistinguishable from zero (not significant), then we ignore the associated variable entirely. To do otherwise would require us to discuss an infinity of possible variables as if they mattered to the model.
> correlation between unemployment and cognitive indicators
If you're arguing that the author should have dropped all observations of unemployed persons from the dataset, that's completely separate and has nothing to do with parabolas.
Your other point, that the parabola could be "not statistically significant," is true.
But given a strong degree of significant correlation between unemployment and the cognitive indicators, even if the dependence is totally flat for the cognitive indicators between 5 hours worked and 100 hours worked, you will still get a parabola by this method of statistical analysis.
Do not forget, this is model fitting.