Linear regression is a general term. When used, $y=ax+b+\epsilon$ is what comes to mind first, however $y=ax^2+bx+c+\epsilon$ is also linear regression, i.e. $x_2=x, x_1=x^2$ and $y=ax_1+bx_2+c+\epsilon$. It's just we use polynomial features. The data (target) can be of parabolic nature but it can still be estimated via linear regression if you use polynomial features. High bias occurs when you use overly simplistic model compared to the data; not specifically when you use the model $y=ax+b+\epsilon$.
Unbiased estimator is a slightly different concept. If an estimator, say $\hat{\theta}$ for a variable $\theta$ is unbiased, then we have $E[\hat{\theta}]=\theta$. A very simple unbiased estimator is the mean; it is also unbiased since if $\hat{\theta}=\mu$ then the expected value of it will equal to the mean: $E[\hat{\theta}]=E[\mu]=\mu=E[\theta]$. Therefore, let alone OLS, using just the mean is an unbiased estimation technique. So, having an unbiased estimator doesn't mean that your estimator fits well to your data.