Can airplane seats have parasites?

Very Unlikely. It is very unlikely that pests such as bedbugs will be transmitted via airline seats. Most parasites are obtained in their 'natural' enviroment: from swimming in infected water, walking barefoot on dirt roads/beaches, mosquito bites, or from drinking/eating contaminiated food and water. Check with your doctor and the cdc website prior to travel for more infromation.
Lice. Lice can be spread by vector transmission for instance the lice can be on brushes, combs, hats, headphones, scarves, sweaters, coats, bedding, towels, upholstered furniture, carpet, stuffed toys & seats in a variety of areas. If you are concerned bring a lint remover with you on the plane.
Not Normally. If flying in the USA, very remote occurrence - more chance in a restaurant. Flying in a remote foreign country - maybe, even their food is a gamble too.