You have a choice - and a further option.
The amount of VAT a Parish Council can reclaim without being registered is quite modest but may be enough for most, most of the time.
However, if you undertake a capital project with substantial VAT on the purchase, you are unlikely to be able to reclaim all the VAT. Registering for VAT would allow you to reclaim all the VAT but does, as suggested, require you to charge VAT on your income.
If you are careful when you register you can opt selected facilities out of VAT so you do not have to charge VAT on those parts. In theory, if you were registering to reclaim VAT on a new sports pavilion, you could register but opt not to tax everything else, so the additional VAT is only charged on use of the pavilion and, you could even protect users from that increased cost by paying it yourself.