Not only does the form or putting names on the title matter; but more so the facts, the substance or actual occurrences, determine the ownership and the tax consequences.
For example, not putting the parents name on the title does not change that a gift was made of the funds to buy the property.
Canada does not have gift tax and US gift tax does not apply to gifts from nonresident aliens.
There is no gift tax in any event from this transaction.
If the parents are simply giving the entire purchase price then only putting you or him (or you and your fiancéé) name on the title is making the form fit the actual facts.
Depending on who the gift is being made to would make the choice for only putting you or him or you and him on the title. Who the gift is made to could have future estate tax consequence.
If the parents intend to keep some ownership (such as to later be paid back when sold) then those facts might be that one half of the price is being gifted and the other half will be owned by them (so any gain would be theirs when sold). If they keep ownership then that property would be in their estates.
Under those facts three or four names - depending on if the gift is being made to you or both you and him would be used.
As to whether it is best to have this titled only in your name, only his name, or in the name of you and your future spouse (congratulations, btw) might depend on estate tax exposure. If it is intended to keep this item out of your or his estate then not having the gift to that party and not having that name on the title may be most proper.
Income tax consequence on sale will be the same whether owned by one or both spouses. There are exclusions from gain for primary residence used two of the five years prior to sale but either spouse can be used to qualify. If the parents keep ownership there will not be exclusion on gain from their share of the home sale.
As stated, the facts behind the transaction determine the tax consequence and the form, or titling, should follow those facts.
Please ask if you need more discussion or clarification.
Different rules apply for nonresident aliens that are domiciled in the US or not. Non-U.S. citizens who are non-U.S. domiciliaries are generally allowed a reduced estate tax exemption amount, which permits only $60,000 of U.S. situs assets to be transferred free of U.S.estate tax.
Having your father make the purchase has created an estate that may later be subject to US estate tax.
Of course, this message is general information and it is understood that you need to engage a practitioner for an opinion and to verify the correct application to your circumstances.
It is possible to avoid any gift tax and some or all of the estate tax by only giving gifts up to the exclusion amount (currently $14,000) each year.
That may be best done with three names on the title (at some point if not in closing) and with annual letters to each of you and your fiancéé gifting that annual amount of value each year. Likely this could include a letter for the first year of ownership.
By limiting the amount given to the annual gift exclusion there is no gift tax return required, no gift tax and that amount is moved out of the father's estate. The amount is indexed for inflation but it will take more than twenty years to transfer all the value from one donor to two recipients if all gifts are below the exclusion amount.
Alternatively, not sure if it was already considered to gift the cash to you and have you make the purchase as cash from a nonresident is not subject to gift tax unlike real property in the US (but it may be too late to do so, depending on the purchase agreement).
Please continue to ask if you need more discussion or clarification.
One important qualifier that has not yet been mentioned is that under the Canada-U.S. Tax Treaty, Canadian residents have a U.S. estate tax liability only if their worldwide assets are valued at more than $5.25 million.
That is, no estate tax will be due even if more than $60,000 of real property is owned in the US unless the total worldwide assets value more than $5.25 million (and that amount also now has inflation indexing possible).
So, the possible estate tax for US property may or may not be a consideration depending on the total worldwide assets of your father.