We present two variants of OMD which are robust against nonce misuse. Security of OMD---a CAESAR candidate---relies on the assumption that implementations always ensure correct use of nonce (a.k.a. message number); namely that, the nonce never gets repeated. However, in some application environments, this non-repetitiveness requirement on nonce might be compromised or ignored, yielding to full collapse of the security guaranty. We aim to reach maximal possible level of robustness against repeated nonces, as defined by Rogaway and Shrimpton (FSE 2006) under the name misuse-resistant AE (MRAE). Our first scheme, called misuse-resistant OMD (MR-OMD), is designed to be substantially similar to OMD while achieving stronger security goals; hence, being able to reuse any existing common code/hardware. Our second scheme, called parallelizable misuse-resistant OMD (PMR-OMD), further deviates from the original OMD design in its encryption process, providing a parallelizable algorithm, in contrast with OMD and MR-OMD which have serial encryption/decryption processes. Both MR-OMD and PMR-OMD are single-key mode of operation. It is known that maximally robust MRAE schemes are necessarily two-pass, a price paid compared to a one-pass scheme such as OMD. Nevertheless, in MR-OMD and PMR-OMD, we combine the two passes in a way that minimizes the incurred additional cost: the overhead incurred by the second pass in our two-pass variants is about 50 % of the encryption time for OMD.