This paper describes and compares two straightforward approaches for dependency parsing with partial annotations (PA). The first approach is based on a forest-based training objective for two CRF parsers, i.e., a biaffine neural network graph-based parser (Biaffine) and a traditional log-linear graph-based parser (LLGPar). The second approach is based on the idea of constrained decoding for three parsers, i.e., a traditional linear graph-based parser (LGPar), a globally normalized neural network transition-based parser (GN3Par) and a traditional linear transition-based parser (LTPar). For the test phase, constrained decoding is also used for completing partial trees. We conduct experiments on Penn Treebank under three different settings for simulating PA, i.e., random, most uncertain, and divergent outputs from the five parsers. The results show that LLGPar is most effective in directly learning from PA, and other parsers can achieve best performance when PAs are completed into full trees by LLGPar.