From what I understand, there's not a way to do that directly (at least not now). Seems like something that should be added. They recently added some 'retweet' capabilities, seem logical to add this as well.
Here's one possible way to do this, first sample tweet data (from status/show):
<status>
<created_at>Tue Apr 07 22:52:51 +0000 2009</created_at>
<id>1472669360</id>
<text>At least I can get your humor through tweets. RT @abdur: I don't mean this in a bad way, but genetically speaking your a cul-de-sac.</text>
<source><a href="http://www.tweetdeck.com/">TweetDeck</a></source>
<truncated>false</truncated>
<in_reply_to_status_id></in_reply_to_status_id>
<in_reply_to_user_id></in_reply_to_user_id>
<favorited>false</favorited>
<in_reply_to_screen_name></in_reply_to_screen_name>
<user>
<id>1401881</id>
...
From status/show you can find the user's id. Then statuses/mentions_timeline will return a list of status for a user. Just parse that return looking for a in_reply_to_status_id matching the original tweet's id.
1) statuses/update returns id of the last status (if include_entities is true)
2) Then you can request statuses/mentions and filter the result by in_reply_to_status_id. The latter should be equal to the particular id from step 1
Twitter has an undocumented api called related_results. It will give you replies for the specified tweet id. Not sure how reliable it is as its experimental, however this is the same api call that is called on twitter web.
It requires you to know the Twitter user's screen_name attribute as well as the id_str attribute of the tweet in question. This way, you can get an arbitrary conversation feed from any arbitrary user's tweet:
*UPDATE: Refreshed code to reflect object access vs array access:
This class should do exactly what you want.
It scrapes the HTML of the mobile site, and parses a conversation. I've used it for a while and it seems very reliable.
Here I am sharing simple R code to fetch reply of specific tweet
userName = "SrBachchan"
##fetch tweets from @userName timeline
tweets = userTimeline(userName,n = 1)
## converting tweets list to DataFrame
tweets <- twListToDF(tweets)
## building queryString to fetch retweets
queryString = paste0("to:",userName)
## retrieving tweet ID for which reply is to be fetched
Id = tweets[1,"id"]
## fetching all the reply to userName
rply = searchTwitter(queryString, sinceID = Id)
rply = twListToDF(rply)
## eliminate all the reply other then reply to required tweet Id
rply = rply[!rply$replyToSID > Id,]
rply = rply[!rply$replyToSID < Id,]
rply = rply[complete.cases(rply[,"replyToSID"]),]
## now rply DataFrame contains all the required replies.
since statuses/mentions_timeline will return the 20 most recent mention this won't be that efficient to call, and it has limitations like 75 requests per window (15min) , insted of this we can use user_timeline
(screen_name== name which we got From status/show)
(count== 1 to max 200)
count: Specifies the number of Tweets to try and retrieve, up to a maximum of 200 per distinct request.
from the result Just parse that return looking for an in_reply_to_status_id matching the original tweet's id.