We introduce the classes IP_R+ (resp. IP_Rx) as the class of languages that admit an interactive protocol on the reals when the verifier is a BSS-machine with addition (resp. addition and multiplication). Let BIP_R+ (resp. BIP_Rx) its restriction when only boolean messages can be exchanged between the prover and the verifier. We prove that the classes BIP_R+ and PAR_R+, the class of languages accepted in parallel polynomial time coincide. In the case of multiplicative machines, we show that BIP_Rx ⊆ PAR_Rx). We also separate BIP _R from IP_R in both models by exhibiting a language L which is not in PAR_Rx but in IP_R+. As a consequence we show that additive quantifier elimination can't be solved in PAR_Rx and that all boolean languages are in IP_R+.