Difference in destination location between pyproj and geopy

It looks like you've done everything correctly. You can evaluate the errors from each method by performing the inverse calculations to find the distance given the origin and destination coordinates, then evaluate the residuals of distances. This is a round-trip exercise.

# For Vincenty's method:
geopy_inv_dist = geopy.distance.vincenty(origin, destination).m
# For Karney's method:
prj_inv_dist = g.inv(long_1, lat_1, prj_long_2, prj_lat_2)[2]  # s12

print("> inverse distance residule (m)")
print("geopy  %.7f" % (distance * 1000 - geopy_inv_dist))
print("prj    %.7f" % (distance * 1000 - prj_inv_dist))

Shows:

> inverse distance residule (m)
geopy  0.1434377
prj    0.0000000

So you can see that Vincenty's method determines an inverse distance that is over a decimetre different for the same coordinates. Karney's method has errors within machine precision, which is less than 15 nanometers. In this example the error is 0.1455 nm, which is around the diameter of a hydrogen atom.

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The problem is probably with geopy's destination method. Let's compare a second implementation of Vincenty's method with PostGIS versions 2.1, shown here. (PostGIS version 2.2 with Proj 4.9 and later use Karney's methods). The round-trip distance residuals from PostGIS 2.1 is always less than 1 cm. For this example it is 255 nm:

SELECT PostGIS_Version(),
  ST_AsText(origin) AS origin,
  ST_AsText(ST_Project(origin, distance, azimuth)) AS destination,
  ST_Distance(ST_Project(origin, distance, azimuth), origin) AS roundtrip_distance,
  distance - ST_Distance(ST_Project(origin, distance, azimuth), origin) AS postgis_residual
FROM (
  SELECT 124.8 * 1000 AS distance, radians(96.021667) AS azimuth,
    ST_SetSRID(ST_MakePoint(-1.729722, 53.320556), 4326)::geography AS origin
) AS f;
-[ RECORD 1 ]------+-----------------------------------------
postgis_version    | 2.1 USE_GEOS=1 USE_PROJ=1 USE_STATS=1
origin             | POINT(-1.729722 53.320556)
destination        | POINT(0.12720134063267 53.1884316458524)
roundtrip_distance | 124799.999999745
postgis_residual   | 2.54993210546672e-007

The geopy pull request fixes your issue with geopy. You will need to install the python package geographiclib first with

pip install geographiclib