Since this is JSON, you probably shouldn't bother trying to get a regex working. Instead, you can parse the JSON and then use LINQ.
using System.Web.Script.Serialization; // (in System.Web.Extensions.dll)
...
string s = "[{\"ta_id\":97497,\"partner_id\":\"229547\",\"partner_url\":\"http://partner.com/deeplink/to/229547\"},{\"ta_id\":97832,\"partner_id\":\"id34234\",\"partner_url\":\"http://partner.com/deeplink/to/id34234\"}]";
JavaScriptSerializer j = new JavaScriptSerializer();
object[] objects = (object[])j.DeserializeObject(s);
string[] ids = objects.Cast<Dictionary<string, object>>()
.Select(dict => (string)dict["partner_id"])
.ToArray();
It's a little messy to deserialize it to an object, because you don't have any type information. If you're not afraid of making a small class to deserialize into, you can do something like this:
class Foo
{
public string partner_id
{
get;
set;
}
}
...
JavaScriptSerializer j = new JavaScriptSerializer();
string[] ids = j.Deserialize<Foo[]>(s).Select(x => x.partner_id).ToArray();
Note that there are other options for deserializing JSON. I simply chose the most general-purpose one that's built in.