Tomato
A low skirt of parsley at the base of staked tomatoes takes up little room, and any plant allowed to open its small flat flower clusters feeds hoverflies and tiny parasitic wasps that hunt aphids. Under low pest pressure this can keep aphid colonies on the tomatoes in check.
Asparagus
Gardeners have long tucked parsley between asparagus crowns, and no mechanism has been demonstrated. It is at least practical: parsley roots stay shallow enough to avoid the crowns, and the bed is left undug for years, which suits a slow starter.
Chives
Carrot fly finds its hosts largely by smell, and the onion scent of a chive clump can help cover the scent of carrot-family foliage nearby. Treat it as a small bonus, not a substitute for insect mesh over young parsley.
Carrot
Old charts keep these two relatives together simply because they want the same loose, evenly moist soil. There is no demonstrated benefit, and they share carrot fly and leaf spot, so mesh both from sowing and move the whole planting together at rotation time.
Rose
Parsley is said to protect roses from aphids, a pairing repeated in cottage-garden writing without supporting trials. It does make a tidy edging that shades the soil at the base of the canes, so treat the planting as ornamental rather than protective.