  We use 47 gravitational-wave sources from the Third LIGO-Virgo-KAGRA
Gravitational-Wave Transient Catalog (GWTC-3) to estimate the Hubble parameter
$H(z)$, including its current value, the Hubble constant $H_0$. Each
gravitational-wave (GW) signal provides the luminosity distance to the source
and we estimate the corresponding redshift using two methods: the redshifted
masses and a galaxy catalog. Using the binary black hole (BBH) redshifted
masses, we simultaneously infer the source mass distribution and $H(z)$. The
source mass distribution displays a peak around $34\, {\rm M_\odot}$, followed
by a drop-off. Assuming this mass scale does not evolve with redshift results
in a $H(z)$ measurement, yielding $H_0=68^{+12}_{-7} {\rm
km\,s^{-1}\,Mpc^{-1}}$ ($68\%$ credible interval) when combined with the $H_0$
measurement from GW170817 and its electromagnetic counterpart. This represents
an improvement of 17% with respect to the $H_0$ estimate from GWTC-1. The
second method associates each GW event with its probable host galaxy in the
catalog GLADE+, statistically marginalizing over the redshifts of each event's
potential hosts. Assuming a fixed BBH population, we estimate a value of
$H_0=68^{+8}_{-6} {\rm km\,s^{-1}\,Mpc^{-1}}$ with the galaxy catalog method,
an improvement of 42% with respect to our GWTC-1 result and 20% with respect to
recent $H_0$ studies using GWTC-2 events. However, we show that this result is
strongly impacted by assumptions about the BBH source mass distribution; the
only event which is not strongly impacted by such assumptions (and is thus
informative about $H_0$) is the well-localized event GW190814.
