Online search shows his parents as Randolph Peyton and Mary Speed.
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LINCOLN COUNTY, KENTUCKY MARRIAGES
Speed Peyton married Zurtha Peyton
12 August 1836
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SEMI-WEEKLY INTERIOR JOURNAL, STANFORD, KY.
TUESDAY, FEBRUARY 12, 1884
Page 1 Hustonville
Died on Friday, at 6 o'clock P. M., Matthew Speed Peyton, aged 71 years. Notwithstanding he attained to an advanced age, Mr. Peyton was from childhood of a delicate constitution. His last illness was protracted and the termination seemed to be simply a wearing out of the system.

FRIDAY, FEBRUARY 15, 1884
Page 2 Obituary
M. Speed Peyton died at his home in Hustonville, Ky., Feb. 8, 1884, aged seventy years, two months and twenty one days. For the satisfaction of his numerous friends and relatives living in different states, I desire to submit the following brief statement:
His wife had preceded him to that "land beyond" the river twelve years. The immediate cause of his death was a lung disease that had been threatening his health for years. His life was marked by prudence, peacefulness, generosity, unquestioned integrity and an exquisite sense of honor. If he had enemies, a sense of his purity paralyzed the tongue of reproach, so that the extremest timerity was never known to charge aught of wrong to his unsullied name. In the esteem of his neighbors he was the very soul of honor. Such fame he fairly won by a life of industry, generosity and christian integrity. Such is the patrimony, more precious than gold or rubies, he bequeathed to ten surviving children, whose sorrowing hearts cleave in unfaltering devotion to his memory. He became a member of the McCormack Christian church about forty years ago and remained such until death. The associations of his earlier life bound his heart to that consecrated place so that his request was that his body should be buried there by the side of his faithful wife. His life was symmetrical, no uncouth angularities marred its harmonious proportions. His faith in the Savior was unwavering. He loved all christian men irrespective of sect and died in the hope of eternal life through the infinite merits of Jesus Christ.
He died without any regrets over a mistaken life which so often cast their mournful shadows over men's last hours on earth. The influence of his life can not be arrested by a grave. That is immortal. "Though dead he yet speaks." May the purity of his christian life be reproduced in the lives of his surviving children. "God be with them 'til we meet again." W. L. W.
(Kentuckiana Digital Library)
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The newspapers show he had a daughter named Julia B. Peyton. I can't find when or where she died.