At low levels, especially in the dunes, there is usually a shortage of white mages, and many will want red mage to be main healer. I personally hated this, and ultimately gave up red mage before I even got out of the jungles. It's very likely the other guy was willing to play the job to fit the needs of the parties - in other words, focus completely on being a healer.
Now, RDM will never be as good as WHM, and this is especially true at the lower levels. RDM just doesn't get access to decent healing spells until a bit later. But when you're in the dunes, something is better than nothing, and so the burden usually falls to RDM or SMN, sometimes even asking BLM to heal if they have a WHM subbed.
My only suggestion is that you change your gear focus away from any sort of melee (sword, scale mail), and focus more on magic attributes. Get sanity rings or eremites, get a wand, make sure you have a justice badge, and the Friar's rope is great at your level. Read up on the RDM forums on this board. There is a lot of debate about whether RDM should melee - personally I am one of those that thinks RDM should not melee in most party situations, but there are always exceptions.
If you don't want to give up melee, and you don't want to be main healer, that is certainly your prerogative, and I don't advocate doing anything you don't want to. But realize that the reason others may get more invites than you is because they may be willing to adapt their play style to the wishes of others. I am by no means saying you have to do this, but if you want more party invites, this could be a big reason why.
Part of going through the dunes is just luck. Somtimes you can get in on great parties, other times you can wait hours on end. I simply suggest trying to solo in LaThienne or the Highlands while waiting for an invite. Slow xp is better than none, and it also gives you the chance to farm a bit - try targeting mobs that have drops you can stack and sell on the AH.
Oh, and as for linkshells, most of the time when you party you will meet a few people you really like (and others you don't , of course). If you are having a good time in a party, ask one of them if they can join their linkshell. They might say no, but usually people are more than willing to hand out pearls there. I'm sure you'd also be welcomed in Fable Unity, since membership is wide open. visit www.fableunity.com to learn about the group (go to message boards) and talk to Sophus or one of the pearlsack holders.
Best of luck to you.
Edited, Fri Oct 14 08:02:10 2005 by lawtechguy