The two books illustrated by Kay Nielsen are:
The Twelve Dancing Princesses and Other Tales for Children retold by SIR ARTHUR QUILLER-COUCH
East of the Sun and West of the Moon: Old Tales from the North
Here are two books illustrated by Maxfield Parrish:
The Arabian Nights, edited by KATE DOUGLAS WIGGIN and NORA A. SMITH
Poems of Childhood, by EUGENE FIELD
Some other books with especially fine illustrations in colour are:
A Child’s Garden of Verses, by ROBERT LOUIS STEVENSON, illustrated by Jessie Willcox Smith
The Wind in the Willows, by KENNETH GRAHAME, illustrated by Nancy Barnhart
LOUIS DODGE’S The Sandman’s Forest and his The Sandman’s Mountain, both illustrated by Paul Bransom
MARY MAPES DODGE’S Hans Brinker, or, The Silver Skates, illustrated by George Wharton Edwards
FRANCES HODGSON BURNETT’S Little Lord Fauntleroy, with illustrations by Reginald B. Birch