I've seen websites placing HTTPS iframes on HTTP pages.

Are there any security concerns with this? Is it secure to transmit private information like credit card details in such a scheme (where the information is only placed on the HTTPS iframe form, and not on the HTTP parent page)?

If only the iframe is https, the user cannot trivially see the URL it points to.  Therefore, the source http page could be altered to point the iframe anywhere it wanted to. That's pretty much a game-over vulnerability that eliminates the advantages of https.