Bismuth ferrite (BiFeO3) is one of the most promising multiferroics with a sufficiently high ferroelectric (FE) and antiferromagnetic transition temperatures, and magnetoelectric (ME) coupling coefficient at room temperature, and thus it is highly sensitive to the impact of cross-influence of applied electric and magnetic fields. According to the urgent demands of nanotechnology miniaturization for ultra-high density data storage in advanced nonvolatile memory cells, it is very important to reduce the sizes of multiferroic nanoparticles in the self-assembled arrays without serious deterioration of their properties. We study size effects of the phase diagrams, FE and ME properties of semi-ellipsoidal BiFeO3 nanoparticles clamped to a rigid conductive substrate. The spatial distribution of the spontaneous polarization vector inside the nanoparticles, phase diagrams and paramagnetoelectric (PME) coefficient were calculated in the framework of modified Landau-Ginzburg-Devonshire (LGD) approach. Analytical expressions were derived for the dependences of the FE transition temperature, average polarization, linear dielectric susceptibility and PME coefficient on the particle sizes for a general case of a semi-ellipsoidal nanoparticles with three different semi-axes a, b and height c. The analyses of the obtained results leads to the conclusion that the size effect of the phase diagrams, spontaneous polarization and PME coefficient is rather sensitive to the particle sizes aspect ratio in the polarization direction, and less sensitive to the absolute values of the sizes per se. ; Comment: 18 pages, 4 figures, to be submitted to Journal of Alloys and Compounds