With unparalleled rotational stability, millisecond pulsars (MSPs) serve as ideal laboratories for numerous astrophysical studies, many of which require precise knowledge of the distance and/or velocity of the MSP. Here, we present the astrometric results for 18 MSPs of the "MSPSR$\pi$" project focusing exclusively on astrometry of MSPs, which includes the re-analysis of 3 previously published sources. On top of a standardized data reduction protocol, more complex strategies (i.e., normal and inverse-referenced 1D interpolation) were employed where possible to further improve astrometric precision. We derived astrometric parameters using sterne, a new Bayesian astrometry inference package that allows the incorporation of prior information based on pulsar timing where applicable. We measured significant ($>3\,\sigma$) parallax-based distances for 15 MSPs, including $0.81\pm0.02\,$kpc for PSR J1518+4904 -- the most significant model-independent distance ever measured for a double neutron star system. For each MSP with a well-constrained distance, we estimated its transverse space velocity and radial acceleration. Among the estimated radial accelerations, the updated ones of PSR J1012+5307 and PSR J1738+0333 impose new constraints on dipole gravitational radiation and the time derivative of Newton's gravitational constant. Additionally, significant angular broadening was detected for PSR J1643-1224, which offers an independent check of the postulated association between the HII region Sh 2-27 and the main scattering screen of PSR J1643-1224. Finally, the upper limit of the death line of $\gamma$-ray-emitting pulsars is refined with the new radial acceleration of the hitherto least energetic $\gamma$-ray pulsar PSR J1730-2304.