OK, the "less demanding" question does seem more tractable; a few possible answers follow, though none is clearly the most "nice pleasant way of parametrizing" your curve. One direction leads to the trigonometric solution of a cubic equation with all roots real; the other leads to an elliptic curve with 6-torsion, and even to an extremal elliptic K3 surface! Which if any of these is best for you is a matter of taste and of what you're trying to do with these curves.

Let $(X,Y,Z) = (\sin^2 \alpha, \phantom.\sin^2 \beta, \phantom.\sin^2 \gamma)$. Then $(X,Y,Z)$ are coordinates of an algebraic curve
$$
E_c : X+Y+Z = c^2, \phantom{=} X^2+Y^2+Z^2 - 2(YZ+ZX+XY) + 4XYZ = 0.
$$
So far we've preserved the $S_3$ symmetry, and can recover the original variables via
$\alpha = \arcsin X^{1/2} = \frac12 \arccos(1-2X)$ and likewise for $\beta,\gamma$.
But this begs the question of what $E_c$ looks like, and leaves us with multivalued arcsines or arccosines. The latter problem seems inherent in another symmetry of the equation: we can translate $\alpha,\beta,\gamma$ by $a\pi,b\pi,c\pi$ for any integers $a,b,c$ with $a+b+c=0$. But we can try to do more with $E_c$.

One direction is to express everything in terms of elementary symmetric functions $\sigma_1,\sigma_2,\sigma_3$ of $X,Y,Z$, as R.Bryant did: the first equation says $\sigma_1=c^2$, and the second says $\sigma_1^2 = 4 (\sigma_3 - \sigma_2)$; so $(\sigma_1,\sigma_2,\sigma_3)$ are parametrized in terms of $\sigma_3$,
and then $X,Y,Z$ are the three roots of
$$
0 = u^3 - \sigma_1 u^2 + \sigma_2 u - \sigma_3 = u^3 - c^2 u^2 + (\sigma_3 + \tfrac14 c^4)u - \sigma_3.
$$
This is still manifestly symmetric but rather implicit. We we can solve the cubic; since it has three real roots the solution will involve trisecting some auxiliary angle $\theta$, itself given as the arccosine of some explicit but complicated algebraic function of $c$ and $\sigma_3$. The roots will then be given in terms of $c$, $\sigma_3$, and the cosines of $\theta/3$, $(\theta+2\pi)/3$, and $(\theta+4\pi)/3$, and the action of $S_3$ will correspond to replacing $\theta$ by the equivalent $\pm (\theta+2\pi n)$ for some $n \bmod 3$. This will be far from nice and pleasant (compare with the formulas for constructing a regular 13-gon using an angle trisector, as in p.192 of Gleason's *Monthly* article), but it will have the advantage of leaving the symmetry close to the surface.

Another direction is to consider $E_c$ on its own terms. It is an elliptic curve, so rational functions on it like $x,y,z$ can be parametrized by elliptic functions like $\wp$ and $\wp'$. Moreover $E_c$ inherits the $S_3$ action so the resulting formulas must retain this symmetry; and the periodicity of $\wp,\wp'$ may even cancel out the ambiguity in the arcsine or arccosine. That's great if you love elliptic curves, not so great if you regard $\wp$ as yet another obscure transcendental function... At least these elliptic curves are rather nice: the cyclic permutations of $X,Y,Z$ are translations by 3-torsion points of $E_c$, and there's also a 2-torsion point because switching two of the variables, say $Y \leftrightarrow Z$, has a rational fixed point where the third variable vanishes (this corresponds to taking $\alpha = 0$ and $\beta+\gamma=0$ in the original equation). So $E_c$ actually has 6-torsion. If I did this right, an equivalent equation for $E_c$ is
$$
y^2 = x^3 + ((c^2-3) x - (c^2-2)^2)^2,
$$
which exhibits the 3-torsion points where $x=0$, and has 2-torsion at $(x,y) = -((c^2-2)^2,0)$.
As it happens $E_c$ is not far from the universal elliptic curve with a 6-torsion point, which is given by $y^2 = x^3 + ((h-3)x - (h-2)^2)^2$. What's more, our substitution $h=c^2$ produces an elliptic K3 surface whose fiber $E_c$ becomes singular at the familiar points $c = 0, \phantom.\pm \sqrt2, \phantom.\pm \frac32$, and also $c=\infty$ — and the multiplicities at $c=0,\phantom.\pm\sqrt2,\phantom.\infty$ are large enough that this elliptic K3 surface is "extremal" (finite Mordell-Weil group, maximal Picard number)! Such surfaces have attracted considerable attention over the years, starting with the Miranda-Persson list of semistable extremal surfaces (*Math. Z.* **201** (1989), 339–361), which includes ours with multiplicity vector $[1,1,4,6,6,6]$. This makes your family of curves very nice in that context, even if it doesn't do much to answer your motivating question...