This article considers an F_q-conic contained in an F_q-subplane of PG(2,q^3), and shows that it corresponds to a normal rational curve in the Bruck-Bose representation in PG(6,q). This article then characterises which normal rational curves of PG(6,q) correspond via the Bruck-Bose representation to F_q-conics of PG(2,q^3). The normal rational curves of interest are called 3-special, which relates to how the extension of the normal rational curve meets the transversal lines of the regular 2-spread of the Bruck-Bose representation. This article uses geometric arguments that exploit the interaction between the Bruck-Bose representation of PG(2,q^3) in PG(6,q), and the Bose representation of PG(2,q^3) in PG(8,q).