Jeannie
By the way, I got a speedy and helpful response from the Southwest Precinct Liaison Attorney. Here are the most relevant parts:
Animal Control is the correct agency to contact about leash enforcement. The Animal Shelter can be reached at 206-386-7387.
– SMC 18.12.080: Animals running at large are prohibited in parks. it is unlawful for any person to allow or permit any dog or other pet to run at large in any park, or to permit any dog or other pet with or without a leash, except Seeing Eye or Hearing Ear dogs or dogs used by public law enforcement agencies and under control of a law enforcement officer, to enter any public beach, swimming or wading area, pond, fountain, stream, organized athletics area or designated children’s play area. The Superintendent may ban dogs and other pets, or a specific dog or other pet, from areas of any park where he or she determines the same may be a nuisance. http://clerk.seattle.gov/~scripts/nph-brs.exe?s1=&s2=leash&S3=&Sect4=AND&l=0&Sect3=PLURON&Sect5=CODE1&d=CODE&p=1&u=%2F%7Epublic%2Fcode1.htm&r=5&Sect6=HITOFF&f=G
Keep in mind, some Seattle parks are off-leash areas. It is not illegal to have a dog off-leash in these areas. Here is a link to off-leash parks: http://www.seattle.gov/parks/offleash.asp.
You may want to contact the Parks Department about your concerns regarding off-leash behavior at Lincoln Park: (206) 684-4075.