Yup, as madcat said, these kinds of posts were made with Rhymezone.
Tumblr Tag: #rhymezone
Rhymezone
Rhymezone is a website that serves up rhyming and nearly-rhyming words and phrases for a searched word or phrase, organized by number of letters or syllables. It's especially helpful for writing poetry and song lyrics. (Or, in this case, swapping out parts of existing poetry and songs with other words while preserving the rhymes/rhythm.)
Rhymezone is not a new site, nor is its use. Rhymezone's help page states it's been in operation since 1996. Google Insights shows a spike around now for the search term Rhymezone, but this spike doesn't majorly stand out compared to its long history.
My research (Google searching Tumblr)
I searched Google for posts on Tumblr relating to Rhymezone.
In Google search results, posts like the ones we're talking about show up as "[song lyrics] image" (The screencapped word/phrase shows up in the search blurb as "image". Example: "all the other kids with the. image. better run, better run. image.").
So I searched site:tumblr.com rhymezone image
Total number of results (all time): 9,340
Number of results by year:
2008 – 0 results
2009 – 3 results
2010 – 6 results
2011 – 9 results
2012 – 389 results
2013* – 235 results
*from January to today (less than 3 months)
So it seems the Rhymezone posts on Tumblr weren't really substantial till sometime in 2012. And we're close to hitting last year's total only a few months into 2013.
I decided to look a little deeper.
Number of results broken down by month from 2012 onward:
Jan: 3
Feb: 4
Mar: 5
Apr: 4
May: 6
Jun: 2
Jul: 4
Aug: 3
Sep: 8
Oct: 5
Nov: 73
Dec: 219
Jan: 152
Feb: 51
Mar*: 60
*to date (Mar 1st to Mar 23rd)
So, more specifically, it started picking up in November 2012 and peaked in December 2012. It's been slowing since then.
The oldest Rhymezone post on Tumblr that I could find via Google is one titled "A poem by rhymezone.com", which reads "roses are red / violets are blue / skillet corn bread / wandering jew". It was originally posted on Apr 14, 2009, according to the timestamp provided by Google, by deactivated Tumblr user wooksie. A reblog of this post can be found here.
Now of course, the info I provided here is just Tumblr stuff since it was the easiest to search through, but I hope this helps shed some light. The next step would be looking outside of Tumblr for similar silly word-swapping.