Based on the pseudocode from @MichaelTodd, and the what you listed above, I cobbled together some code that I think will work.
In essence, it deals with the ID's that contain a type of "A" first.
1. Select those ID's
2. Select features matching those ID's.
3. Delete features where Type <> A
Next, deal with the features that don't contain a Type A. This should now only include features that were not in the first set, since those have been removed.
1. Select ID's where Type <>A
2. Step through ID's, selecting features with matching attribute
3. Count features matching each ID, then iterate through deleting features until only 1 is left.
arcpy.env.workspace = r"c:\Temp"
fc = "newshape.shp"
field1 = "Site_ID"
field2 = "Site_Type"
field3 = "OID"
#Setting delimiters should allow you to change source without causing problem in query strings
delimfield1 = arcpy.AddFieldDelimiters(fc,field1)
delimfield2 = arcpy.AddFieldDelimiters(fc,field2)
delimfield3 = arcpy.AddFieldDelimiters(fc,field3)
srchstring = delimfield2 + "= 'A'"
#Returns features with Site_Type = A
rowswitha = arcpy.SearchCursor(fc,srchstring,"",field1, "Site_ID A")
#Makes list of Site_ID's that have one or more feature where Site_Type = A
idswitha = 
for itemwitha in rowswitha:
#Step through list of Features, delete features in ID list where Site_ID <> A
for idwitha in idswitha:
idstring = delimfield1 + " = " + idwitha
rowswithid = arcpy.UpdateCursor(fc,idstring)
for rowwithid in rowswithid:
del rowswithid, rowwithid
#Create list of features where Site_Type not equal to A
srchstring2 = delimfield2 + " <> 'A'"
fields = field3 + "; " + field1
rowsnoa = arcpy.SearchCursor(fc,srchstring2,"",fields, "Site_ID A")
#Create list of ID's where Wite_Type <>A
idsnoa = 
for itemnoa in rowsnoa:
#Create Unique value list of IDs
setidsnoa = set(idsnoa)
#For each ID, return features, count features, then delete until deleted count leaves 1
for idnoa in idsnoa:
idnoastring = delimfield1 + " = " + idnoa
rowsnoaid = arcpy.UpdateCursor(fc, idnoastring)
#Create list to return count of items in cursor
rowcountlist = 
for rownoaid in rowsnoaid:
#Get count of items in list
rowcount = len(rowcountlist)
count = 0
rowdel = rowsnoaid.next()
while count < rowcount:
rowdel = rownoaid.next()
I edited to add list item to return count for second section of the query. It is not the most efficient code, so any improvements are welcome.
I removed the index call from UpdateCursor as this is not supported. Changed to