Mean estimate and Least square estimate. This question is refers to the parameter estimate by average value given by the link:
https://en.wikipedia.org/wiki/Mean
and least square estimates by the link:
https://en.wikipedia.org/?title=Least_squares
The both link talks about parameter estimation. Can we compare the both? It appears to me if we want to estimate two independent parameters then least square estimates becomes the mean estimates? Is it true? Can we prove it somehow?

No, the mean only happens to be the least squares parameter estimate for a select few models. In general, you are calculating something more complicated. We can demonstrate this using maximum likelihood methods:
Let $f(x;\alpha,\beta)=g(x;\alpha)h(x;\beta)$, so that the estimates for $\alpha$ and $\beta$ are independent. Then the log-likelihood of a sample,$y=(y_1,y_2,...y_n)$ is:
$$\mathcal{L}(\alpha,\beta;y)=\sum_y\log(g(y_i;\alpha))+\sum_y\log(h(y_i;\beta))$$
We maximize the log-likelihood by setting its derivatives to 0 (and checking that it is concave). These will be our maximum likelihood estimates $(\hat{\alpha},\hat{\beta})$:
$$\hat{\alpha} = b: \frac{\partial \mathcal{L}}{\partial \alpha}\mid_{\alpha=b}=0 \implies \sum_y\frac{dg(y_i;\alpha)}{d\alpha}|_b\frac{1}{g(y_i;\alpha)}=0$$
And similarly for $\beta$. 
Note that neither of these imply the mean is the estimator nor that least squares is the appropriate method.
