We used the OSIRIS camera at the 10.4m Gran Telescopio Canarias (GTC) to monitor the astrometric motion of the L4.5 dwarf 2M1821+14 over 17 months. The astrometric residuals of 11 epochs have an rms dispersion of 0.4mas, which is larger than the average precision of 0.23mas per epoch and hints towards an additional signal or excess noise. Comparison of the point-spread functions in OSIRIS and FORS2/VLT images reveals no differences critical for high-precision astrometry, despite the GTC's segmented primary mirror. We attribute the excess noise to an unknown effect that may be uncovered with additional data. For 2M1821+14, we measured a relative parallax of 10^6.15^+/-0.18mas and determined a correction of 0.50+/-0.05mas to absolute parallax, leading to a distance of 9.38+/-0.03pc. We excluded at 3{sigma} confidence the presence of a companion to 2M1821+14 down to a mass ratio of 0.1 (~5M_Jupiter_) with a period of 50-1000d and a separation of 0.1-0.7au. The accurate parallax allowed us to estimate the age and mass of 2M1821+14 of 120-700Myr and 0.049^+0.014^_-0.024_M_{sun}_, thus confirming its intermediate age and substellar mass. We complement our study with a parallax and proper motion catalogue of 587 stars (i'=~15.5-22) close to 2M1821+14, used as astrometric references. This study demonstrates submas astrometry with the GTC, a capability applicable for a variety of science cases including the search for extrasolar planets and relevant for future astrometric observations with E-ELT and TMT.