Which orbital in a subshell gets filled first?

Let's say we're filling electrons in subshell 2p. This subshell will have $m_l = -1$, $m_l = 0$, and $m_l = 1$ orbitals. Does the 'filling' of electrons depend on the value of $m_l$? Meaning, does the value of $m_l$ determine if it gets filled first by an electron or not?

The only really important thing when filling electrons into lone atoms in vacuum is taking care to fill shells from lowest to highest principal quantum number $n$ and the azimuthal quantum number $l$, since those are the only two that affect the electron’s energy.
$n$ affects even the energies of electrons in hydrogen-like atoms while $l$ only comes into play once core electrons exist whose orbital shapes can make a difference to the overall charge distribution. This all makes macrophysical sense, since $n$ is directly part of the quantitised electron energy $E_n$ and $l$ is the quantum number determining the electron’s angular momentum with $l(l+1)\hbar$ being the eigenvalue of the squared angular momentum operator $\hat{\vec{L}^2}$.
The orientation of this (or more precisely: its $z$ component, $\hat{L_z}$ by its eigenvalue $m_l \hbar$) is determined by the magnetic quantum number your question is asking about. Stick to the image of a single atom in vacuum: There is no reason why certain directions of angular momentum (remember the value is the same, only the orientation changes) should in any way affect the energy — especially since $z$ is a completely arbitrary choice of coordinate; vacuum does not know up or down. It is not until you add surroundings, either by another atom approaching your atom in vacuum or by an external magnetic field that the orientation matters in any way.