The way I visualize this is:
1. The two points on the earth from which the measurements are taken are on opposite sides of the earth from each other.
2. The moon and these two points form an isosceles triangle. The base (which goes between these to measurement points and through the center of the earth) is 1.276*107 m, and the angle opposite the base is 1° 54'.3. What we want to find is the height of this triangle -- the distance from the moon to the center of the earth.
Starting with this triangle, draw the line from the moon to the center of the earth. This line bisects the base, so the moon, one of the measurement points, and the center of the earth form a right triangle. The angle at the point at the moon is 1/2 the original parallax angle. That angle was 1° 54' = 1 54/60° = 1.9°. So the angle at the moon in the right triangle is half that -- 0.95°.
The tangent of this angle equals the base of the right triangle, which is half the diameter of the earth -- 1.276*107 / 2 = 0.648*107 m -- divided by the distance from the moon to the center of the earth. Call that distance h, and we get the following equation:
0.648*107 / h = tan 0.95°
0.648*107 / tan 0.95° = h
390,781,712 m = h
That's the distance from the center of the earth to the moon.
Now, maybe the distance from the earth to the moon should be measured from the SURFACE of the earth, not the center. If so, then you have to subtract the radius of the earth, which we already found to be 0.648*107 m. That gives 384,301,713 m.