I'm afraid tracing to 1799 may not take you back far enough to determine where in England your Parkers came from. You need to establish a line of descent from a member of one of major English/Irish Parkers to have a connection to that person's coat of arms.
In strictest terms, there is no such thing as a family coat of arms. The coat of arms was granted to the individual. The sons, when they earned the right to apply for their own coats of arms, would almost certainly retain an aspect of the father's arms as a sign of respect and of family heritage. There are also cases of a young man marrying a woman who is sole heir to a family estate, with the man adopting a coat of arms based on that of his father in law; for this to happen, there had to be express permission, or the College of Arms would have rejected the application.
From your descriptions, you have found valid coats of arms pertaining to several different Parker families.
The stag's head on the shield with black center band and silver flank bands, a raised arm above grasping an antler, and the motto "fideli certa merces" ("the reward of the faithful is sure"), was first awarded to a Parker from Devonshire, who is the ancestor of the Earls of Morley.
The stag's head on a shield, almost identical to Devonshire arms, but with the motto "Try", appears to belong to the Parkers of Tipperary, which is in Ireland. This is very likely an offshot of the Devonshire Parkers, but I don't know the exact connection.
A shield with a red-brown chevron dividing three leopard heads, each with an arrow in the mouth, and a crest above the shield of a stag pierced by an arrow downward, belonged to the Parkers of Extwistle. Extwistle Hall, located near Worsthorne in Lancashire, is abandoned and derelict, and I do not know if any of that line survive, but the fact that the references to this coat of arms I have come across all refer to it in the past tense lead me to think the Extwistle line is extinct.
A shield with a green chevron dividing three golden stag heads, and a stag in stride above as crest, with the motto "Non fluctu nee flutre movetur" ("unmoved by either wave or wind") is that of the Parkers of Browsholme Hall, also in Lancashire.
A shield with a red-brown chevron dividing three golden leopard faces, a crest of a golden leopard head topped with a ducal crown, and the motto "sepre andre" ("dare to be just"), belongs to the Parker family Norton Lees. Norton Lees is currently a suburb of the city of Sheffield. This coat of arms has descended through the Park Hall and Staffordshire Parker lines, and is currently used by the Earls of Macclesfield.
A shield with a buck in stride between three silver phoens belongs to the Parkers of Cambridgeshire. If there is a crest or motto associated with these arms, I have found no record of it.
There are also the Parkers of Woodthorpe (York), whose arms are described as: "Woodthorpe (York)--Ar. a chevron pean. betw. three mullets sa. on a chief ar. as many bucks' heads caboshed ar. Crest, a talbot's head couped ar. ears and
I'm sure this is an incomplete list, but at least it should give you some things to look for, and perhaps suggest some possibilities concerning geography. You do need to keep in mind that a coat of arms does not establish your ancestry. Only establishing your ancestry can determine your right to a particular coat of arms.