As a teenager, you were probably used to splitting the bill or chipping in with a few friends to help take someone out for their birthday. While this is common for the young, once you're an adult it's considered poor etiquette to invite others to a party, especially a sit-down dinner, and then expect them to pay for it. If you're planning on a more low-key event like a night at the pub, then your friends typically won't expect you to pay the whole bill. It really depends on the type of event you're having.
How to Have a Party on a Budget
When you invite guests to a party for a family member, it's expected that you will be picking up the bill. Therefore, you have two choices if you can't afford to pay for all of your guests. You can reduce the size of your guest list and only invite a few close friends and family - as many as you can afford. Talk to the restaurant in advance to see whether they can present a limited number of affordable menu choices to your guests so that you can stretch your budget. Alternatively, simplify the party and invite everyone to your home for a piece of birthday cake and coffee. Save the fancy restaurant meal for just the two of you.
When It's Ok to Ask Others to Pay
There are exceptions to every rule, of course, as long as all the members of your party are notified in advance that they will be paying for their portion of the tab I think they will be understanding and sympathetic to the situation. If you have a feeling certain friends and family won't agree with this kind of party, simply don't invite them this time. Curate a group that makes sense for this kind of occasion. Alternatively, when birthday parties are held in locations like a local bar or pub it is usually assumed that everyone will pay their own bill. As long as the location is a place where party-goers can drop in and out then having people cover their own bill won't seem rude. If you are planning to go to the type of place that will only seat you if your full party has arrived, then asking your friends to pay their share of the bill would seem rude.