What function is used to graph the potential curve of two hydrogens with parallel spins? TL; DR
I am asking for

an analytic potential curve function for the triplet state of $\ce{H2}$ (preferred) with experimental parameters, or
experimental data (enough points to realise a smooth-looking curve).

Also see solid red curve pictured below.

To characterise the energy of a two-atom system, Morse potential is applied as a second approximation after the harmonic oscillator model. This is an analytic function of the form $$E=D_\mathrm{e}\left\{1-\exp[-a(r-r_\mathrm{c})]\right\}$$
where the parameters $D_e$, $a$ and $r_c$ depend on the atoms in play. Surely enough, better results are achievable. For examples on simple molecules, I found the paper by Hulburt and Hirschfelder, 'Potential Energy Functions for Diatomic Molecules', J. Phys. Chem., ($1940$) to be rather useful. Have a look at equations $(14)$ and ($15$) specifically (available more easily here). The one I used was

$$E_{\mathrm{P}}=\mathrm{D_e}[(1-\mathrm{e}^{-z})^2+\mathrm{c}x^3\mathrm{e}^{-2z}(1+\mathrm{b}z)],$$
where
$$z=\frac{\omega_\mathrm{e}}{{2}\sqrt{\mathrm{B_eD_e}}}\left(\frac{x-\mathrm{r_0}}{\mathrm{r_0}}\right)$$

This is for the case where spins are antiparallel.
If atoms of parallel spin approach one another, the qualitative picture is quite different. A smaller distance will equal higher energy regardless of separation, i.e., no stable equilibrium state exists. Unfortunately, finding a nice function for the antibonding situation proved less fruitful. An example of what I'm after is the solid red curve found in this Wolfram demonstration.
I believe that you wont be able to find a closed analytic expression for $\mathrm{H_2}$, since no closed analytic expresion is possible for a many-body problem (more than 2 bodies). That said you can calculate as many points on the triplet curve as you like by using a QM program able of doing unrestricted Hartree-Fock (UHF). Here you should be aware that UHF does not give very good potential curves, because the Hartree-Fock approximation lacks the electron correleation energy. $\mathrm{H_2}$ is a quite small system, so you will be able to make a potential curve using unrestricted MP2 or unrectriced CCSD. I do believe that PySCF is capable of doing such calculations. Regarding the experimental data, I am not aware of any databases for such kind of data. I would suggest finding a paper where a curve is presented and then use engauge digitizer to extract the points you need.