Marigolds are alleopathic, meaning they emit a protective compound designed by nature to protect the plant itself. Their roots release the chemical alpha-terthienyl, a highly toxic naturally occurring compound. This compound helps to control nematodes, some insects and plant pathogens. French and Mexican Marigolds produce a much stronger version of this chemical. The Mexican ones have the potential to hinder the growth of other plants, in particular some of the more tender herbs. Once planted, the alpha-terthienyl accumulates in the soil and will protect for years to come. Don't expect immediate results as the compound accumulates over time, usually by the second year of planting marigolds in the same plot you will see the improvement.
Petunias deter and repel asparagus beetles, as well as many leaf-hoppers, aphids, and tomato hornworms. They work well with Asparagus.
Calendula - Repels nematodes in the soil, but attracts slugs.
Chrysanthemum - Repels nematodes in the soil. Possibly acts as a deterrent to other pests, although it is of limited value on some asparagus pests.
Herbs such as parsley, basil, coriander and dill will deter some pests, spider mites, aphids and so forth. Basil deters tomato hornworms. When inter-planting with these herbs and flowers, be aware that although they are compatible with asparagus, they may not be compatible with each other.
Carrots will do well with Asparagus, they are a close relative of parsley and act as a deterrent of some asparagus pests. Carrots do compete somewhat with asparagus for nutrients which do not make them an ideal candidate - but they are workable and somewhat compatible.
Tomatoes help to protect asparagus against asparagus beetles. Tomatoes produce a substance called solanine which protects against the Asparagus beetle. Asparagus produces an amino acid "asparagine", which is harmful to many tomato plant pests. A spray derived from asparagus is effective on tomato plants as a killer of nematodes, including the root-knot sting, stubby root and meadow varieties. Tomatoes and Asparagus are a true symbiotic relationship.
"A good method for planting [Tomato and Asparagus] is in a long row at one side of the garden. After harvest, plant tomatoes on either side, and both plants reap benefits from each other. Parsley planted with asparagus seems toprovide vigor to both "- Cornell University - Companion Planting
Contrary to popular belief there is no real reason to avoid planting Potatoes near Asparagus. Like tomatoes, potatoes produce solanine which repels asparagus pests. Other than competition for nutrients, which isn't a factor so long as you don't plant them right on top of each other - there is no reason Potatoes and Asparagus can't be planted in proximity to each other.
Planting Potatoes and Asparagus - Martha Stewart
Comfrey is very deep rooted plant that acts as a dynamic accumulator, it accumulates various minerals and nutrients from deep within the soil. This attribute makes it available not only to the comfrey, but neighboring plants as well. Comfrey has rapidly growing foliage which breaks down quickly and adds to the surrounding soil composition. It supplies many of the nutrients needed by Asparagus. However - In my opinion- I don't believe Comfrey makes a great companion plant for Asparagus. Comfrey gets huge, very huge and over-bearing it easily crowds out Asparagus and would require constant maintenance. On a pro and con scale - the cons far outweigh the pros when it comes to planting Comfrey and Asparagus together.
Strawberries are sometimes companion planted with Asparagus. This is fine for strictly aesthetic purposes. But the Strawberry does nothing for the Asparagus and vice versa. In addition, they are not really even compatible as their pH requirements aren't even close. If you are so inclined - planting strawberries and asparagus together is workable, but there are many other options that are both pleasing to the eye as well as edilicously productive.Companion Planting: Strawberries, Asparagus, Rhubarb and Horseradish - Mother Earth News