Even if source
parameter is not supported, you can quickly create you own utility to get what you want, e.g.
In[1]:= twitter = ServiceConnect["Twitter", "New"]
Out[1]= ServiceObject["Twitter",
"ID" -> "connection-29f751249df0b3e87a365f8c21a6b31f"]
In[20]:= url =
"https://api.twitter.com/1.1/statuses/user_timeline.json";
In[21]:= userid = Lookup[First@twitter["UserData"], "ID"];
(*build url with required paramaters*)
In[30]:= urlstring =
URLBuild[{url}, {"user_id" -> ToString@userid,
"screen_name" -> "realDonaldTrump", "count" -> 5}];
(*fetch data using authentication framework*)
In[31]:= result = OAuthClient`oauthdata[twitter, urlstring];
(*post processing result*)
In[33]:= data = ImportString[result[[2]], "RawJSON"];
In[34]:= #["source"] & /@ data
Out[34]= {
"<a href=\"http://twitter.com/download/iphone\" \rel=\"nofollow\">Twitter for iPhone</a>",
"<a href=\"http://twitter.com/download/iphone\" rel=\"nofollow\">Twitter \for iPhone</a>",
"<a href=\"http://twitter.com/download/iphone\" rel=\"nofollow\">Twitter for iPhone</a>",
"<a href=\"http://twitter.com/download/iphone\" rel=\"nofollow\">Twitter \for iPhone</a>",
"<a href=\"http://twitter.com/download/iphone\" rel=\"nofollow\">Twitter for iPhone</a>"
}