I conducted a polynomial regression of the form $y(x) = A+Bx + Bx^2$, and would like to obtain parameters $A$, $B$ and $C$ and their respective errors/confidence level. The reduced chi square is less than 1, suggesting that errors might be overestimated. I got advised to multiply the $y$ errors with the square root of the reduced chi square so to reduce the size of the error and obtain a reduced chisquare equal to 1. \begin{equation} \sigma_{y_{i}new} = \sqrt{\tilde{\chi_{0}}} \cdot\sigma_{y_{i}} \end{equation} Does this make sense? And if it does, why is it so? I don't really understand if this is something that really can be done or if it's too much manipulation on the data.
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This is not an uncommon thing to do, and it is ok-ish so long as you are transparent about how you estimated the error bars in any publications/presentations. Roughly speaking you are "re-calibrating" your estimated error bars, with the observed uncertainty due to the goodness-of-fit coming from the $\chi^2$. This approach is not unreasonable if you trust your model describe the data well, and you aren't very confident in your error bars.
To be honest I usually see this applied in the other direction -- in other words, more often than not, the reduced $\chi^2$ is larger than one, and people move in the direction of being more conservative by increasing ("inflating") their error bars. But, it's not really worse to apply this method in the opposite direction, so long as you are transparent.
The big caveat is here is that your model may be overfitting the data. If that is happening, then decreasing your error bars is hiding a problem under the rug. A small $\chi^2$ can indicate that your model has too many parameters and is learning some spurious patterns from the noise. Since your model is still fairly simple (just a quadratic), this probably isn't too big a problem for you, although impossible to say for sure without looking at the details. Another question I would have, is whether you have a good theoretical justification for fitting a quadratic function to the data, or else how did you determine to use a quadratic function instead of a linear or cubic one. A systematic way to check for overfitting is to use part (say 80%) of your dataset to fit the model (the "training set"), and then compute the chi^2 of the model against the other 20% of your data (the "test set") -- if you find the reduced $\chi^2$ value is a lot lower on the training set than the test set, this is a sign your model is overfitting.
tl;dr: Without more context it's hard to give an absolute answer. So long as you are transparent, and especially since the polynomial is a pretty low order one, this approach can be fine and viewed as a quick-and-dirty way to correct for a lack of first-principles understanding of the error bars. But there is a risk of overfitting if you aren't careful with this type of approach.
