A set of criteria was developed for finding short-period comets among those for which only parabolic orbits are available. These criteria indicate that the basic Catalogue of Cometary Orbits by Marsden (1989) includes about 18 such cases with revolution periods less than 20 years. About 15 of them were observed before the discovery of the first known object of this type. One of them, comet 1678, was already definitely identified with P/d'Arrest (Carusi et al., 1991). Another 14 of them - comets 1230, 1345, 1351, 1457 I, 1491 II, 1499, 1539, 1577 II, 1585, 1702, 1743 I, 1833, 1949 III, and 1963 IX - were almost certainly of short period (p(s) = 95 to 99%). For comets 568, 1080 and 1457 II a short revolution period appears probable (p(s) = 80%), for comets 1245, 1293, 1618 I and 1618 III possible (p(s) = 50%), and for comets 390, 1757 and 1860 IV not impossible (p(s), = 20%). There are some cases of orbital similarity with the currently known short-period comets (P/Finlay, P/Denning-Fujikawa, P/Biela, P/d'Arrest, P/Barnard 3 and P/Denning), which do not exclude the possibility that they may be earlier apparitions of these comets. Any unambiguous identification, however, will be very difficult without historical records of additional returns. At the same time, it would be very important for our understanding of the physical lifetimes of comets, their long-term dynamical evolution, and of the variable nongravitational forces acting on them.