We introduce the class IP_ℛ+ (resp. IP_ℛ×) 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_ℛ×) be 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_ℛ× ⊆ PAR_ℛ×. We also separate BIP_ℛ from IP_ℛ in both models by exhibiting a language L which is not in PAR_ℛ× but in IP_ℛ×. As a consequence we show that additive quantifier elimination cannot be solved in PAR_ℛ× and that all boolean languages admit an interactive proof with addition and a real constant.