Is it OK to extend furnace gas pipe to the other side of the house (traversal) to connect a barbeque?
"Is such an extension allowed?"

That's entirely up to at least two entities that "the internet" has no particular way of knowing about, but you do. Call them.


Your gas company (and that may involve an assesment of whether the
furnace pipe is of adequate size to feed both appliances, or not)
Your "Local Authority Having Jurisdiction" or LAHJ - town/city/county/state may have a dim view of DIY gas piping, or they may merely require that your gas company bless it before commissioning, or they may not give a fig - though the last is uncommon in LAHJs with piped gas. Your gas supplier may be helpful in telling you who you need to call locally.