as far as I know the Countesthorpe records aren't available online though I could be wrong.The Leic FHS produces a CD or download which contains details of Poor Law admissions, including to the Countesthopre homes - you would have to buy this from them, via their webpage.
I can see various admissions on the files which are probably your family, but Parker is not an uncommon name, the ages are not exact, and the children are shown as Church of England on occasion and Chapel on others, which makes me wonder if I'm looking at the right ones.
Having said that, here's a brief summary of admission dates to the workhouse (from where children were transferred to the Cottage Homes) and comments where made:
Arthur - adm 23 Jan 1892, age 1, deserted, chapel
Arthur - adm 9 April 1892, age 0, mother Fanny Parker, C of E
Eliza - 9 April 1887, age 9, mother Fanny Parker, CofE
Eliza - 23 Jan 1892, age 13, desertion, chapel
Harriet - 19 Jan 1887, age 5, cof E, mother Fanny Parker
Harriet - 23 Jan 1892 with Eliza and Arthur
9 April 1892 with Arthur
then in and out of the workhouse over the next 10 years
William (also called Willie) - admitted with the others in Nov 1887 age 3, Jan 1992, April 1992, transferred back from the Cottage Home in Jun 1892
Fanny - Nov 1887 (age 40) and Jan and April 1992 (age 30!) with the children. There was a death for Fanny Parker in the workhouse on 16 Sept 1892, this may well have been her.
There are several admissions for adults called William Parker but without knowing his dob it's difficult to know which might be him.