From Heritage Quest -- probably your folks:
1860 census -- Warren, Lake Co IL
William Mildrum, 44, Scot
Jane Mildrum, 32, Scot
Anne Mildrum, 9, Scot
Jane Mildrum, 7, Scot
Elise Mildrum, 5, IL
William Mildrum, 2, IL
Alick Mildrum, 8/12, IL
Meldrum is one of my main lines. My ggg-grandparents were Rev. William Meldrum (1765-1854) & Mary McDonald (1773-1838). There were from County Tyrone, Northern Ireland -- arrived in USA about 1802 -- settled in KY for 30 years then relocated to Greene Co., IL (about 60 miles North of St. Louis). The Clan probably originated in the Aberdeen Scotland area a few more generations back.
Rootsweb also has a Meldrum Surname Message Board. Have you tried that?