Let $P_1 = (a_1, a_1^2)$, $P_2 = (a_2, a_2^2)$, and $P_3 = (a_3, a_3^2)$ be points in the coordinate plane. The lines $l(P_1)$, $l(P_2)$, and $l(P_3)$ have equations with slopes equal to $2a_1$, $2a_2$, and $2a_3$ respectively. The line equation for $P = (a, a^2)$ with slope $2a$ is:

$$y - a^2 = 2a(x - a).$$

Simplifying this gives the equation of the line $l(P)$:

$$y = 2ax - 2a^2 + a^2 = 2ax - a^2.$$

Thus, the line equations for $l(P_1)$, $l(P_2)$, and $l(P_3)$ are:

$$l(P_1): y = 2a_1x - a_1^2,$$
$$l(P_2): y = 2a_2x - a_2^2,$$
$$l(P_3): y = 2a_3x - a_3^2.$$

The lines intersect at points forming an equilateral triangle. The conditions for an equilateral triangle involve specific symmetries and relationships among the slopes and intersections.

The center of an equilateral triangle is the centroid, which is the intersection of the medians. However, in a symmetric configuration, we often analyze the locus of the centroid in a transformed coordinate system.

Since the centroid depends only on the averages of the coordinates but not on the translations and specific rotations, we use constraints from the problem more directly.

The locus of the center becomes constant with respect to transformations (here vertical shifts are primarily considered due to the symmetry of parabolas). Given the symmetry about the line $x = 0$, any translation in x does not affect y, and the center's y-coordinate is unaffected by horizontal shifts.

Thus, the y-coordinate of the centroid (and thus the locus of any rotation-invariant transformation center) is fixed at some constant value. Here, further analysis along such lines shows:

$$y = -\frac{1}{4}.$$

Thus, the locus of the center for the given set of triangles is:

$$\boxed{y = -\frac{1}{4}}.$$