What tape is best to use?
I wrapped several grafts with a 1/2” parafilm tape to prevent from drying and all of them decomposed because of the moisture excess inside.
It seems the parafilm doesn’t let the air to get through and the wood is suffocating. I wrapped the unions and the scions.

Hola Igor my friend, Don't know about the parafilm. Could it be other factors that killed it, Heat, it did not have good contact, not tight enough or weak cutting. I had a lot of loss of outdoor grafts due to some extreme heat in March.

I've had good results with Parafilm.

Agree with Armando,I lost many also, not to parafin, as I never used it. It is a mistery to me, how some glue to the stalk,others not.  I have a calvert that was grafted onto a 1" thick Adriatic, it was gorgeous last fall,  this spring, something (likely gnats and their friends) ate 98% of the Adriatic bark, but not where the Calvert was attaching... now I have a totally bare stalk with a fig growing out of  it... I better get some pictures soon.  Others are fine.  It is mostly cold and rainy here, but we are having a very unusual sunny spring into summer...

it is a long trial and error learning process. I wish you don't give up as you have done amazing work with your little figs. 

Hmm.. So you say it is something else? 
I did a surgical inspection on one very promising graft and found that it had healed nicely and was still attached to the stem but the area covered by the Parafilm was soft on touch and started to rot. I was keeping the fig location dry on purpose, no water was getting in, and it was in a shade, no direct sun.
I think I will try a different approach this weekend. I will dip the scion on the candle wax and wrap with the Parafilm the union only.
In 2 weeks it is going to be clear if it took or not.

Igor, I've used BuddyTape (a form of Parafilm) with success and haven't experienced that sort of trouble at all.  Parafilm does, in fact, let air in but traps the larger particles of water in.  Unless the union was "bleeding", I don't see where you would be getting a source of moisture from.

Did you stretch the Parafilm as you applied it?  I've seen some people do it incorrectly and not stretch it.  In fact, I did that the first couple of times I used it since I had not seen this explained.  Without being stretched it might not let air into the graft area.

I've also used Nescofilm, a superior product (IMO) which is no longer manufactured.  An old information sheet on it compared the oxygen exchange rate of it to Parafilm).

I worry about mold on my grafts. I leave them on for only about 30 days, probably wrong to remove wrapping so soon.....I have been using small green branches (scion) lately, it shows me quicker results. If it doesn't adhere well to the branch, it shrivels up in about a week. I of course remove it immediately avoiding potential mold.

The grafts I did were not green, they were some of the cuttings I collected over the past winter. They are nice and viable but this tape gives me the trouble (or whatever I am doing wrong).

When you applied the parafilm was it stretched tight?  The only way I can imagine that would happen if the film wasn't stretched tight and sealing.  Someone...Tim I think...wraps his entire above-soil cuttings in parafilm to prevent dessication. It takes practice to get it stretched tight but not break, warming it in your hands as you wrap helps a TON.

The grafts I did on my pears showed no condensation or anything like that...they just didn't do well because of my terrible technique. Now I know better for next year!

Kelby,
I believe the parafilm was tight but since I removed the failed grafts, I cannot go back and check. It is possible that was not tight enough.

Will the dipping the scions into the candle wax (paraffin) instead of the parafilm do the trick?
It seems to be easier that way, the scions will be ready to go and I need to work on the union only.

I stretch my parafilm to near-breaking.  It thins and stretches quite a bit. 

I end up with several layers over the graft as I wrap around again and again.  The rest of the scion just gets one or two layers.  I leave the tape on till it falls off.  On the grafted area, that happens pretty quickly, I think because there is a lot of expansion of the callous. 

The rest of the scion tends to remain wrapped for months.  Most of my grafts from March/April still have the parafilm on them away from the grafted area.  They do tend to get mildewed, but the grafts don't seem to suffer for it.

I have a tree that had it's one inch trunk gnawed down to half an inch by a squirrel a few months ago. I wrapped it tightly in Buddy Tape. The tree has since been thriving and has fruit on it. I removed the tape this morning and found that the bottom inch of the six inches I wrapped was covered with what I think are root initials. The bark that was under tape looks just as healthy as the bark that was not wrapped. The injury seems to have callused over well. No sign of rotting or moisture buildup. 

I also find root initials, and more than initials, under the parafilm occasionally.  Usually right around the graft area, but sometimes along either the also-wrapped ajacent rootstock and/or scion.  They shrivel pretty quickly once the tape is removed.

It is very important to keep your grafts in the SHADE. If they are in direct sun, they will get cooked. If the rootstock is a larger in-ground tree, cover the graft with white paper, or some other means of shading the scion material. Even if they don't get cooked, the increased heat can lead to mold, drying out, and other problems.

I have never tried grafting and I'm getting ready to try it for the first time this winter. So one thing that keeps coming up is parafilm. haha.

Is this how you do it more or less. Or does it need to be a lot tighter?

Here is another video where they us hydrofilm, are these the correct level of tightness? (start video around 5:15, thats when they apply the hydrofilm to the graft)

I've been wrapping my cuttings as tight as I can without ripping the tape. Are you keeping them in a humid area after they have been wrapped? I can't think how they would be rotting for you otherwise.

I also wrap mine as tight as I can. No problems here. I am actually amazed how the plant just pushes through the tape.

If you are having problems stretching the Parafilm, it might be an older roll.  I have trouble after a few years with that.

That might also explain why after it has been out in the elements the plant doesn't have much problem growing through it.

They use the buddy tape so fast!!! OMG!!!

In the middle video I see two unions on each stick that look like two grafts. What are those?

I cant remember exaclty what video on YouTube explains it but Tom Spelman eluded to it once, my memory is failing me, but he called it a new technique that floyd ziager was having great success with

1) bottom stock type --Super vigorous
2) intermediate root stock--middle
3) the top scion -- the variety you want is on top and you've given it the best the best components

Interesting, I learned something today!
So we can try to whip and tongue graft x fig scions, that would be really cool!
I can get tons of local figs to play with. The question is, should I root first the rootstock or graft them before rooting.
It seems a rooted rootstock may have a higher chance to support the scions.
And I can try the green tape Scott was referring to also.