Bhakti and Sufi movements were the two prominent strands of socio-religious movements going on in medieval India. Both were transforming how the devotee worshiped the divine. Both of them also worked to reduce the religious tension in society, preached social equality, and opposed pomp and orthodoxy.
Both these movements were operating in a caste-divided, orthodox, and multi-religious society. Hence, they often prescribed the same kinds of reforms.
Both Bhaktism and Sufism had a well-developed body of literature that departed from the orthodox doctrines of Hinduism and Islam respectively. Both propagated love to the god as their central tenet. Sufism promoted syncretism and decried the orthodoxy of Ulemas. Bhaktism encouraged the participation of lower castes and women in bhajan-kirtan. Bhakti saints talked about universal brotherhood and gave voice to newly emerging classes and castes, e.g. Guru Nanak quickly gained a large group of followers among Jat peasantry. Sufi saints consciously choose themes from Indian mythology, music, and local language to promote harmony between Islam and Hinduism. Sufis opposed pomp and ostentation just as popular Bhakti saints like Mira, Namdev, etc. Kabir repeatedly pointed out religious hypocrisy in both religions and promoted moral behavior in place of ritual fastidiousness. But, both these movements had numerous sub-strands; they should not be called parallel movements in respective religions.
Thus, both Bhaktism and Sufism promoted conciliation, social reforms and simplicity in society and religion.