Can the decoder in a transformer model be parallelized like the encoder?
The correct answer is: computation in a Transformer decoder can be parallelized during training, but not during actual translation (or, in a wider sense, generating output sequences for new input sequences during a testing phase).
What exactly is parallelized?
Also, it's worth mentioning that "parallelization" in this case means to compute encoder or decoder states in paralllel for all positions of the input sequence. Parallelization over several layers is not possible: the first layer of a multi-layer encoder or decoder still needs to finish computing all positions in parallel before the second layer can start computing.
Why can the decoder be parallelized position-wise during training?
For each position in the input sequence, a Transformer decoder produces a decoder state as an output. (The decoder state is then used to eventually predict a token in the target sequence.)
In order to compute one decoder state for a particular position in the sequence of states, the network consumes as inputs: 1) the entire input sequence and 2) the target words that were generated previously.
During training, the target words generated previously are known, since they are taken from the target side of our parallel training data. This is the reason why computation can be factored over positions.
During inference (also called "testing", or "translation"), the target words previously generated are predicted by the model, and computing decoder states must be performed sequentially for this reason.
Comparison to RNN models
While Transformers can parallelize over input positions during training, an encoder-decoder model based on RNNs cannot parallelize positions. This means that Transformers are generally faster to train, while RNNs are faster for inference.
This observation leads to the nowadays common practice of training Transformer models and then using sequence-level distillation to learn an RNN model that mimicks the trained Transformer, for faster inference.