At my last session one of the characters ended up being paralyzed- the creature stat block read as follows:

 it (the target) must succeed on a DC 13 Constitution saving throw or be poisoned taking 18 (4d8) poison damage on a failed save, or half as much damage on a successful one. In addition, on a failed save the target is paralyzed for 1 hour. (This is a homebrew creature I found on dndbeyond- a werespider, from what I can tell it is the same kind of paralysis you would get from a giant spider) 

After a few rounds of being paralyzed the player asked if there was a saving throw they can make to end the paralysis. I saw nothing in the creatures stat block nor could I find anything in the rules, so I said no, sorry (and unfortunately it was the cleric that was paralyzed). I am wondering if that is truly the case- is there no rule on making a saving throw to end paralysis? Thanks guys.  

Paralysis, like most conditions, doesn't specify it's own duration or causing effect. This is always put into the information by whatever it is that causes the condition. For Example, the Web spell specifies how a creature can end the restrained condition that it afflicts:

"Each creature that starts its turn in the webs or that enters them during its turn must make a Dexterity saving throw. On a failed save, the creature is restrained as long as it remains in the webs or until it breaks free. A creature restrained by the webs can use its action to make a Strength check against your spell save DC. If it succeeds, it is no longer restrained."

Looking at a Giant Spider (the paralysis source for the homebrew creature)

"Bite. Melee Weapon Attack: +5 to hit, reach 5 ft., one creature. Hit: 7 (1d8 + 3) piercing damage, and the target must make a DC 11 Constitution saving throw, taking 9 (2d8) poison damage on a failed save, or half as much damage on a successful one. If the poison damage reduces the target to 0 hit points, the target is stable but poisoned for 1 hour, even after regaining hit points, and is paralyzed while poisoned in this way." 

The paralysis is linked to the poison, but only kicks in if the poison reduces the target to 0 hit points. Poisoned also doesn't have it's own additional saves (in the case of the Giant Spider, once you are not poisoned, you are not paralyzed).

According to your quote from the homebrew creature, on the failed save, the target is paralyzed for one hour and doesn't get to make additional saves. Compare this to the Chuul, which does specify that you can repeat the saving throw at the end of each turn.

"Tentacles. One creature grappled by the chuul must succeed on a DC 13 Constitution saving throw or be poisoned for 1 minute. Until this poison ends, the target is paralyzed. The target can repeat the saving throw at the end of each of its turns, ending the effect on itself on a success."

it doesn't say they can remae the saving throw in the stat block, so i'm pretty sure he's just paralyzed until the hour passes or a spell that ends the condition is cast on him

I would say given the duration seems to be from the same paralysis source as a Giant Spider, it may be a mistake for the HomeBrew Creature to apply 1 hour paralysis (generally this is only present like with Giant Spider (when the Poison drops the player to 0hp)) on the single failed save.

I've had a quick look, I believe you are using this homebrew Werespider? In this case, I'd 100% rework the poison to be the same as for a Giant Spider because it looks like they've decided to make it a one time save paralysis.

This 100% is not a balanced CR5 creature, there is no other paralysis in 5e that works the way this Werespider is designed. Paralysis from a single save, either allows repeated saves at the end of each turn, ends at the end of the next of the afflicting creatures turn or, only triggers when the poison that is linked to it, drops the creature to 0hp.

Unfortunately, usually anything that causes paralysis also describes how it can end. For example, the hold person spell allows a saving throw that the end of each of your turns. In this case, the description simply says that on a failed save the creature is paralyzed for an hour. Since you are the DM, you can change it up if you want but there is no general rule on ending paralysis (though the spell lesser restoration can also end the condition).

Yes, I agree that it is not balanced, I am actually making my own version now. 

I'd say you can fix it easily by adding any of the mechanisms used by non-homebrew creatures that can apply paralysis

Okay, created my own version. 

https://www.dndbeyond.com/monsters/1212305-werespider-drow

Looks good, I like the change to lycanthropic curse on a bite with poison on the sting, makes more sense that way