The search and study of close pairs of supermassive black holes (SMBHs) is important in the study of galaxy mergers that can possibly trigger active galactic nucleus (AGN) activity, and in the context of their evolution into the gravitational wave emitting regime. The quasar SDSS J1048+0055 was identified as an SMBH binary (SMBHB) candidate based on the observed double-peaked [O III]lambda lambda 4959, 5007 emission lines and two distinct radio components separated by similar to 20 pc (Zhou et al. 2004). To ascertain the binary nature of this source, we analyzed multifrequency, multiepoch very long baseline interferometry (VLBI) data to investigate its parsec-scale radio properties. The source shows double components with the western feature being brighter than the eastern one. This brighter component has a brightness temperature of similar to 10(10) K, a spectral index of alpha = -0.09 +/- 0.09 (flat), and is indicative of mildly relativistic beaming. In contrast, the faint component has a lower brightness temperature of similar to 10(8-9) K and a steep spectrum. These clues are consistent with a core-jet structure; moreover, the apparent separation speed between the two components is much higher than the expected orbital motion in a binary SMBH. Thus the present study excludes the association of the two VLBI components with the cores of an SMBHB; though, the SMBHB possibility (e.g., a pair of radio-loud and radio-quiet AGNs) is not fully ruled out. In the single AGN scenario, the double-peaked optical emission lines can originate from the jet interacting with the narrow-line region as indicated by a change in the jet direction at similar to 140 pc.