TYPO3 Core

In database data is like that:
<h2 align="center">My Header</h2>

On the site it is transformed into:

{p} <h2 align="center">My Header</h2> {/p}

It is wrapped with {p} {/p}
tags.

{} = <>

If <h2> is htmlspecialchared from DB to website this is not an htmlarea issue. The default-typoscript templates have "<h2>" in the set of allowd HTML-Tags that will not be touched.

However if you or an extension modify the default-setup then this might happen.

plz use the typoScript Object browser and find out about your setting of:
'lib.parseFunc.allowTags'

I don't consider this to be a bug at the moment.

In my typoScript Object browser there is no entry for lib.parseFunc.allowTags.
But when I searched for parseFunc system found a lot of entries and most of them had allowTags property somewhere inside, ie:
plugin.tx_macinabanners_pi1.parser.tags.typolist.1.parseFunc.allowTags = b,i,u,a,img,br,div,center,pre,font,hr,sub,sup,p,strong,em,li,ul,ol,blockquote,strike,span
plugin.tt_news.general_stdWrap.parseFunc.tags.typolist.1.parseFunc.allowTags = b,i,u,a,img,br,div,center,pre,font,hr,sub,sup,p,strong,em,li,ul,ol,blockquote,strike,span
tt_content.table.20.1.parseFunc.allowTags = b,i,u,a,img,br,div,center,pre,font,hr,sub,sup,p,strong,em,li,ul,ol,blockquote,strike,span

and many others identical entries.

I added this lib.parseFunc.allowTags manually (and included h1,h2, h3... in the list) but it didn't change the behaviour.

seems like you are using the old (deprecated) static template "content (default)", typoscript is organized a little bit different then.

however this is just a matter of proper setup/usage. plz consult ML, documentation or any professional for appropriate support.