Retrieving Subkey from a cookie via asp:CookieParameter

I have a cookie that contains two subkeys. I am trying to read one of
the subkeys via the <asp:CookieParameter>, but I can't seem to specify
which subkey to get. ASP.NET seems to only want to retrieve the first
subkey. Does anyone know a workaround?

<asp:CookieParameter CookieName="hospital" Name="hospid"
Type="Int32" />

Code that sets cookie:
HttpCookie strCookie = new HttpCookie("hospital");
strCookie.Values.Add("name", strHospital);
strCookie.Values.Add("id", id.ToString());

Thank you!