The trend component refers to parallel shifts in the term structure, the tilt refers to changes in the shape of the term structure at the long and short ends, and the curvature refers to movements in the medium term part. The phrase 'parallel component' has no meaning and is not a part of the principal components in analyzing term structures.
Changes in the term structure can also be analyzed as "level, slope and curvature", so you should be aware of this terminology as well to refer to the principal components of a term structure analysis.
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