We study the design of good coding schemes and the achievable exchange rates for the parallel Gaussian bidirectional relay channel. We first consider the corresponding linear deterministic model and propose two different schemes that can achieve the exchange capacity for this linear deterministic model. The insights obtained from this are used to design coding schemes for the original parallel Gaussian bi-directional relay channel. The first coding scheme uses superposition-based coding at both nodes and reorders codewords that cannot be transmitted within their own sub-channels to the sub-channels that can support the transmission at the relay. The second coding scheme employs lattice partition chains proposed by Nam et al. in the multiple access phase and then performs coding across sub-channels in the broadcast phase. While both schemes are optimal for the linear deterministic model, the performance of their Gaussian counterparts are different in general and which one performs better depends on the operating SNR and channel coefficients. Numerical results show that both schemes substantially outperform the decode-and-forward scheme and also provide non-trivial gains over the scheme proposed by Huang et al. Moreover, it is shown that the performance of both schemes is close to that of the cut-set bound and that the second scheme is asymptotically optimal.