Confusion with parallelism in verb tenses

Parallelism or a lack of it is often a matter of style rather than grammar (i.e. acceptability). In particular, coordination of verb phrases whose tenses are different is not rare in published literature. Your sentence

[1] Rural hospitals across the US face financial losses and have struggled to stay open.

would generally be judged as acceptable by native speakers.

So would be

[2] Rural hospitals across the US have struggled to stay open and face financial losses.

Whether [1] and/or [2] are lacking in stylistic merit is a separate question. To my ear, they are actually fine. But that is a matter of personal opinion and taste, not English grammar.

Here are some examples from the Corpus of Contemporary American English (COCA)

The Predators don't have terrific center depth and have struggled scoring this season.

The 4,000 strong regional protection force adds to the more than 12,000 peacekeepers who are already in South Sudan and have struggled to protect civilians.

Several of the middle schools there are less than 60 percent full and have struggled to meet state standards.

And here are some related examples from the British National Corpus (BNC)

Most of the twenty or so men and women in the Cabinet are well known to one another and have worked together over some years.

I am pleased to accept the appointment and have taken note of the terms laid down in your letter.

Many LEAs deliberately try to avoid invoking the formal legal procedures unless absolutely necessary and have developed procedures which enable individual truancy cases to be examined 'in the round'.