How to align equations in two columns?

Simple, just use the [t] option for the aligned construct.

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\begin{equation*}
\begin{aligned}[t]
\frac{\partial l(\mu, \sigma|\mathbf{y})}{\partial  \mu} & = -\frac{1}      {2\sigma^{2}}\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}(y_{ij}-\mu)=0 \\
 & = -\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}y_{ij}+n\mu=0\\
 & \Rightarrow \sum_{i=1}^{a}\sum_{j=1}^{n_{i}}y_{ij}=n\mu\\
 & \Rightarrow     \tilde{\mu}=\frac{\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}y_{ij}}{n}=\bar{y}_{..} \\
\end{aligned}
\qquad 
\begin{aligned}[t]
\frac{\partial l(\mu, \sigma|\mathbf{y})}{\partial  \sigma^{2}} &     =\frac{-n}{2\sigma^{2}}+\frac{1}   {2(\sigma^{2})^{2}}\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}(y_{ij}-\mu)^{2}=0 \\ 
& \Rightarrow    \tilde{\sigma}^{2}=\frac{\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}(y_{ij}-\mu)^{2}}{n}  \\ 
& \Rightarrow   \tilde{\sigma}^{2}=\frac{\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}(y_{ij}-  \bar{y}_{..})^{2}}{n}
\end{aligned}
\end{equation*}

\end{document}

What you want, as I understand it, is best obtained with alignat environment, from amsmath:

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\begin{alignat*}{2}
  \frac{\partial l(\mu, \sigma|\mathbf{y})}{\partial \mu} & = -\frac{1} {2\sigma^{2}} \sum_{i=1}^{a}\sum_{j=1}^{n_{i}}(y_{ij}-\mu)=0
  & \qquad\qquad \frac{\partial l(\mu, \sigma|\mathbf{y})}{\partial \sigma^{2}} & =\frac{-n}{2\sigma^{2}}+\frac{1} {2(\sigma^{2})^{2}}\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}(y_{ij}-\mu)^{2}=0 \\
  & = -\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}y_{ij}+n\mu=0
  & & \Rightarrow \tilde{\sigma}^{2}=\frac{\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}(y_{ij}-\mu)^{2}}{n}\\
    & \Rightarrow \sum_{i=1}^{a}\sum_{j=1}^{n_{i}}y_{ij}=n\mu & & \Rightarrow \tilde{\sigma}^{2}=\frac{\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}(y_{ij}- \bar{y}_{..})^{2}}{n} \\
  & \Rightarrow \tilde{\mu}=\frac{\sum_{i=1}^{a}\sum_{j=1}^{n_{i}}y_{ij}}{n}=\bar{y}_{..} \\
\end{alignat*}

\end{document}